Splitting formulas for the rational lift of the Kontsevich integral

TL;DR

This paper extends the Kricker invariant of knots in homology 3-spheres to a functorial setting and proves splitting formulas for it under null Lagrangian-preserving surgeries, generalizing previous results.

Contribution

It introduces a functorial extension of the Kricker invariant and establishes splitting formulas for this extension under a broader class of surgeries.

Findings

Proves splitting formulas for the extended invariant.

Generalizes null-move splitting formulas to null Lagrangian-preserving surgeries.

Enhances understanding of knot invariants in homology 3-spheres.

Abstract

Kricker defined an invariant of knots in homology 3-spheres which is a rational lift of the Kontsevich integral, and proved with Garoufalidis that this invariant satisfies splitting formulas with respect to a surgery move called null-move. We define a functorial extension of the Kricker invariant and prove splitting formulas for this functorial invariant with respect to null Lagrangian-preserving surgery, a generalization of the null-move. We apply these splitting formulas to the Kricker invariant.

Equations

$$M(\mathbf{C})=\left(M\setminus ({\bigsqcup }_{1\le i\le n}{C}_i)\right){\cup }_{\partial }({\bigsqcup }_{1\le i\le n}{C}_i^{\prime }).$$

$$\mu (\mathbf{C})\in \mathrm{hom}({\Lambda }^3{H}^1(\mathcal{C};\mathbb{Q}),\mathbb{Q})\cong {\Lambda }^3{H}_1(\mathcal{C};\mathbb{Q})$$

$$\mu (\mathbf{C})=\mu ({\mathbf{C}}_1)\otimes \cdots \otimes \mu ({\mathbf{C}}_n)\in {\otimes }_{i=1}^n{\Lambda }^3{H}_1({\mathcal{C}}_i;\mathbb{Q}).$$

$${\ell }_{(S,\kappa )}(\mathbf{C}):H_1(\mathcal{C};\mathbb{Q})\times H_1(\mathcal{C};\mathbb{Q})\to \mathbb{Q}(t),$$

$${\ell }_{(S,\kappa )}(\mathbf{C})(a,b)={\text{lk}}_e(\tilde{\alpha },\tilde{\beta }),$$

$${\mathcal{A}}_{\mathbb{Q}}(V)=\frac{\mathbb{Q}⟨V\text{–colored Jacobi diagrams}⟩}{\mathbb{Q}⟨\text{AS, IHX, LV}⟩},$$

$$\begin{array}[]{c c c}S^{n}\Lambda^{3}V&\to&\mathcal{A}_{\mathbb{Q}}(V)\\ (u_{1}\wedge v_{1}\wedge w_{1})\dots(u_{n}\wedge v_{n}\wedge w_{n})&\mapsto&\raisebox{-14.22636pt}{ \leavevmode\hbox to55.88pt{\vbox to33.09pt{\pgfpicture\makeatletter\hbox{\hskip 28.36397pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}{{}} {}{} {}{}{}{{}}{}{{}}{}{{}}{} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }{}\pgfsys@moveto{-19.91684pt}{19.91684pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{21.90865pt}\pgfsys@moveto{0.0pt}{19.91684pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{19.91684pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ 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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\end{array}$$

$${\tilde{\mathcal{A}}}_{\mathbb{Q}(t)}(\varnothing )=\frac{\mathbb{Q}⟨\mathbb{Q}(t)\text{–beaded Jacobi diagrams}⟩}{\mathbb{Q}⟨\text{AS, IHX, LE, Hol, OR}⟩},$$

$$\sum_{I\subset\{1,\dots,n\}}(-1)^{|I|}\tilde{Z}\left((S,\kappa)(\mathbf{C}_{I})\right)\equiv_{n}\left(\textrm{\begin{minipage}{170.71652pt} \begin{center} sum of all ways of gluing all legs of $\mu(\mathbf{C})$ with $\ell_{(S,\kappa)}(\mathbf{C})/2$\end{center}\end{minipage}}\right),$$

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$$\tilde{Z}\left((S^3,\mathcal{O})(\mathbf{C})\right)+\tilde{Z}\left((S^3,\mathcal{O})\right){\equiv }_2-\text{leavevmode}to43.31pt\text{vbox}to48.16pt\text{pgfpicture}\text{makeatletter}\text{lower}-24.18098ptto0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{nullfont}to0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setdash}3.0pt,3.0pt0.0pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@curveto}0.0pt17.07182pt34.14365pt17.07182pt34.14365pt0.0pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setdash}3.0pt,3.0pt0.0pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@curveto}0.0pt-17.07182pt34.14365pt-17.07182pt34.14365pt0.0pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setdash}3.0pt,3.0pt0.0pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@lineto}34.14365pt0.0pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-1.25pt-1.1111pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\bullet \text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.032.89365pt-1.1111pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\bullet \text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@moveto}16.645pt-12.80386pt\text{pgfsys@lineto}16.61183pt-12.80386pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.016.61183pt-12.80386pt\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.015.32182pt-20.84798pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}1\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@moveto}16.645pt0.0pt\text{pgfsys@lineto}16.61183pt0.0pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.016.61183pt0.0pt\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.015.32182pt-8.04411pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}1\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@moveto}16.645pt12.80386pt\text{pgfsys@lineto}16.61183pt12.80386pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.016.61183pt12.80386pt\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.015.80794pt16.33687pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}t\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{pgfsys@discardpath}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{lxSVG@closescope}\text{endpgfpicture},$$

$$0\to H_2(N(K),N(K)\cap E;\mathbb{Q})\to H_1(E;\mathbb{Q})\to H_1(M;\mathbb{Q})\to 0.$$

$$H_1(E;\mathbb{Q})=H_K\oplus (e_{-}{)}_{\ast }(A_{g^{-}})\oplus (e_{+}{)}_{\ast }(B_{g^{+}}),$$

$$H_1(C;\mathbb{Q})\stackrel{inc{l}_{\ast }}{to31.29802pt\text{rightarrowfill}}{H}_1(M\setminus K;\mathbb{Q})⟶H_K$$

$$\mathcal{A}(X,{\ast }_C)=\frac{\mathbb{Q}⟨\text{Jacobi diagrams on }(X,C)⟩}{\mathbb{Q}⟨\text{AS, IHX, STU}⟩},$$

$${\tilde{\mathcal{A}}}_R(X,{\ast }_C)=\frac{\mathbb{Q}⟨R\text{–beaded Jacobi diagrams on }(X,C)⟩}{\mathbb{Q}⟨\text{AS, IHX, STU, LE, OR, Hol}⟩},$$

$${\tilde{\mathcal{A}}}_R^{\text{w}}(X,{\ast }_C)=\frac{\mathbb{Q}⟨R\text{–winding Jacobi diagrams on }(X,C)⟩}{\mathbb{Q}⟨\text{AS, IHX, STU, LE, OR, Hol, Holw}⟩},$$

$${\chi }_S:\bar{\mathcal{A}}(X,{\ast }_{C\cup S})\stackrel{\cong }{to31.29802pt\text{rightarrowfill}}\bar{\mathcal{A}}(X\cup \text{leavevmode}to0.4pt\text{vbox}to11.78pt\text{pgfpicture}\text{makeatletter}\text{lower}-0.2ptto0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{nullfont}to0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@lineto}0.0pt10.92093pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}0.01.0-1.00.00.0pt10.92093pt\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{pgfsys@discardpath}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{lxSVG@closescope}\text{endpgfpicture}{}_S,{\ast }_C).$$

$${\chi }_S:\mathcal{A}(X,{\ast }_C,{\text{*⃝}}_S)\stackrel{\cong }{to31.29802pt\text{rightarrowfill}}\mathcal{A}(X\cup \text{leavevmode}to9.51pt\text{vbox}to9.51pt\text{pgfpicture}\text{makeatletter}\text{lower}-4.75253ptto0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{nullfont}to0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@moveto}4.55254pt0.0pt\text{pgfsys@curveto}4.55254pt2.51433pt2.51433pt4.55254pt0.0pt4.55254pt\text{pgfsys@curveto}-2.51433pt4.55254pt-4.55254pt2.51433pt-4.55254pt0.0pt\text{pgfsys@curveto}-4.55254pt-2.51433pt-2.51433pt-4.55254pt0.0pt-4.55254pt\text{pgfsys@curveto}2.51433pt-4.55254pt4.55254pt-2.51433pt4.55254pt0.0pt\text{pgfsys@closepath}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@moveto}4.55254pt0.85355pt\text{pgfsys@lineto}4.55254pt0.67792pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}0.01.0-1.00.04.55254pt0.67792pt\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{pgfsys@discardpath}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{lxSVG@closescope}\text{endpgfpicture}{}_S,{\ast }_C),$$

$${\chi }_S:{\tilde{\mathcal{A}}}_R(X,{\ast }_C,{\text{*⃝}}_S)\stackrel{\cong }{to31.29802pt\text{rightarrowfill}}{\tilde{\mathcal{A}}}_R(X\cup \text{leavevmode}to9.51pt\text{vbox}to9.51pt\text{pgfpicture}\text{makeatletter}\text{lower}-4.75253ptto0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{nullfont}to0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@moveto}4.55254pt0.0pt\text{pgfsys@curveto}4.55254pt2.51433pt2.51433pt4.55254pt0.0pt4.55254pt\text{pgfsys@curveto}-2.51433pt4.55254pt-4.55254pt2.51433pt-4.55254pt0.0pt\text{pgfsys@curveto}-4.55254pt-2.51433pt-2.51433pt-4.55254pt0.0pt-4.55254pt\text{pgfsys@curveto}2.51433pt-4.55254pt4.55254pt-2.51433pt4.55254pt0.0pt\text{pgfsys@closepath}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@moveto}4.55254pt0.85355pt\text{pgfsys@lineto}4.55254pt0.67792pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}0.01.0-1.00.04.55254pt0.67792pt\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{pgfsys@discardpath}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{lxSVG@closescope}\text{endpgfpicture}{}_S,{\ast }_C),$$

$${\chi }_S:{\tilde{\mathcal{A}}}_R^{\text{w}}(X,{\ast }_C,{\text{*◯⃝}}_S)\stackrel{\cong }{to31.29802pt\text{rightarrowfill}}{\tilde{\mathcal{A}}}_R^{\text{w}}(X\cup \text{leavevmode}to9.51pt\text{vbox}to9.51pt\text{pgfpicture}\text{makeatletter}\text{lower}-4.75253ptto0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{nullfont}to0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@moveto}4.55254pt0.0pt\text{pgfsys@curveto}4.55254pt2.51433pt2.51433pt4.55254pt0.0pt4.55254pt\text{pgfsys@curveto}-2.51433pt4.55254pt-4.55254pt2.51433pt-4.55254pt0.0pt\text{pgfsys@curveto}-4.55254pt-2.51433pt-2.51433pt-4.55254pt0.0pt-4.55254pt\text{pgfsys@curveto}2.51433pt-4.55254pt4.55254pt-2.51433pt4.55254pt0.0pt\text{pgfsys@closepath}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@moveto}4.55254pt0.85355pt\text{pgfsys@lineto}4.55254pt0.67792pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}0.01.0-1.00.04.55254pt0.67792pt\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{pgfsys@discardpath}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{lxSVG@closescope}\text{endpgfpicture}{}_S,{\ast }_C).$$

$$\Delta (D)=\sum _{J\subset I}{D}_J\otimes D_{I\setminus J}.$$

$$\text{leavevmode}to56.77pt\text{vbox}to76.96pt\text{pgfpicture}\text{makeatletter}\text{lower}-12.7271ptto0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{nullfont}to0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setdash}3.0pt,3.0pt0.0pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt25.6073pt\text{pgfsys@lineto}0.0pt13.26364pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}0.0-1.01.00.00.0pt13.26364pt\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.03.533pt8.82603pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}{t}^2\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setdash}3.0pt,3.0pt0.0pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt12.80365pt\text{pgfsys@lineto}0.0pt0.0pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setdash}3.0pt,3.0pt0.0pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt25.6073pt\text{pgfsys@curveto}7.07135pt25.6073pt12.80365pt31.3396pt12.80365pt38.41095pt\text{pgfsys@curveto}12.80365pt45.4823pt7.07135pt51.2146pt0.45999pt51.2146pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}-1.00.00.0-1.00.45999pt51.2146pt\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setdash}3.0pt,3.0pt0.0pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt51.21461pt\text{pgfsys@curveto}-7.07135pt51.21461pt-12.80365pt45.48232pt-12.80365pt38.41096pt\text{pgfsys@curveto}-12.80365pt31.33961pt-7.07135pt25.60732pt0.0pt25.60732pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-3.11374pt-9.39409pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}{s}^{\prime 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$${\int }_{S}G=⟨{\text{exp}}_{\bigsqcup }(-\frac{1}{2}{W}^{-1}(t)),H{⟩}_S\in {\tilde{\mathcal{A}}}_{\mathbb{Q}(t)}^{\text{w}}(X,{\ast }_C).$$

$$w(\gamma_{i},\gamma_{j})=\left\{\begin{array}[]{l l}\displaystyle\frac{1}{2}\sum_{c}\textrm{sg}(c)t^{\varepsilon_{ij}(c)}&\textrm{ if }i\neq j\\ &\\ \displaystyle\frac{1}{2}\sum_{c}\textrm{sg}(c)(t^{\varepsilon_{ii}(c)}+t^{-\varepsilon_{ii}(c)})&\textrm{ if }i=j\end{array}\right.$$

$$W_{\gamma }=W_{\eta }-W_{\eta L}{W}_L^{-1}{W}_{L\eta }.$$

$$W_{{\eta }^{\prime }}=W_{\eta }+{}^{t}P{W}_{L\eta }+W_{\eta L}P+{}^{t}P{W}_{L}P\text{and}{W}_{L{\eta }^{\prime }}=W_{L\eta }+W_{L}P,$$

$$\text{lk}({\xi }^{\prime },{\zeta }^{\prime })=\text{lk}(\xi ,\zeta )-\text{Lk}(\xi ,L).\text{Lk}(L{)}^{-1}.\text{Lk}(L,\zeta ).$$

$${\text{lk}}_e(\zeta ,\xi )=\frac{1}{P(t)}\sum _{j\in \mathbb{Z}}⟨\Sigma ,{\tau }^j(\xi )⟩t^j\in \mathbb{Q}(t).$$

$$w({\gamma }_i,{\gamma }_j)$$

Full text

MSC: 57M27 57M25 57N10

Keywords: 3-manifold, knot, homology sphere, cobordism category, Lagrangian cobordism, bottom-top tangle, beaded Jacobi diagram, Kontsevich integral, LMO invariant, Kricker invariant, Århus integral, Lagrangian-preserving surgery, finite type invariant, splitting formula.

1. 1 Introduction

2. 1.1 Context

3. 1.2 Statement of the main result

4. 1.3 Strategy

5. 2 Domain categories: cobordisms and tangles

6. 2.1 Cobordisms with paths

7. 2.2 Bottom-top tangles with paths

8. 2.3 Tangles with paths and tangles with disks

9. 3 Target categories: Jacobi diagrams with beads

10. 3.1 Diagram spaces

11. 3.2 Product and coproduct

12. 3.3 Formal Gaussian integration

13. 3.4 Categories of diagrams

14. 4 Winding matrices

15. 4.1 First definition

16. 4.2 Topological interpretation

17. 5 Construction of an LMO functor on $\widetilde{\mathcal{LC}ob}$

18. 5.1 The functor $Z^{\bullet}:\widetilde{\mathcal{T}}_{q}\to\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}[t^{\pm 1}]}$

19. 5.2 The functor $Z:\widetilde{\mathcal{T}}_{q}\mathcal{C}ub\to\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}(t)}$

20. 5.3 The functor $\tilde{Z}:\widetilde{\mathcal{LC}ob}_{q}\to{}^{ts}\widetilde{\mathcal{A}}$

21. 5.4 Application to $\mathbb{Q}$SK–pairs

22. 6 Splitting formulas

1 Introduction

1.1 Context

This paper presents the construction of a functorial extension of the Kricker rational lift of the Kontsevich integral, which aims at expliciting the properties of this invariant as a series of finite type invariants.

The notion of finite type invariants first appeared in independent works of Goussarov and Vassiliev involving invariants of knots in the 3–dimensional sphere $S^{3}$; in this case, finite type invariants are also called Vassiliev invariants. Finite type invariants of knots in $S^{3}$ are defined by their polynomial behaviour with respect to crossing changes. The discovery of the Kontsevich integral, which is a universal invariant among all finite type invariants of knots in $S^{3}$, revealed that this class of invariants is very prolific. It is known, for instance, that it dominates all Witten-Reshetikhin-Turaev’s quantum invariants. A theory of finite type invariants can be defined for any kind of topological objects provided that an elementary move on the set of these objects is fixed; the finite type invariants are defined by their polynomial behaviour with respect to this elementary move. For 3–dimensional manifolds, the notion of finite type invariants was introduced by Ohtsuki [Oht96], who constructed the first examples for integral homology 3–spheres, and it has been widely developed and generalized since then. In particular, Goussarov and Habiro independently developed a theory which involves any 3–dimensional manifolds —and their knots— and which contains the Ohtsuki theory for $\mathbb{Z}$–spheres [GGP01, Hab00].

In [Kri00], Kricker constructed a rational lift of the Kontsevich integral of knots in integral homology 3–spheres ($\mathbb{Z}$–spheres). In [GK04], he proved with Garoufalidis that his construction provides an invariant of knots in $\mathbb{Z}$–spheres. They also proved that the Kricker invariant satisfies some splitting formulas with respect to the so-called null-move. For knots in $\mathbb{Z}$–spheres with trivial Alexander polynomial, these formulas together with work of Garoufalidis and Rozansky [GR04] imply that the Kricker invariant is a universal finite type invariant with respect to the null-move.

Kricker’s construction easily generalizes to null-homologous knots in rational homology 3–spheres ($\mathbb{Q}$–spheres); the main goal of this article is to prove splitting formulas for the Kricker invariant of these knots with respect to null Lagrangian-preserving surgery, a move which generalizes the null-move. For null-homologous knots in $\mathbb{Q}$–spheres with trivial Alexander polynomial, these formulas are used in [Mou17] to prove that this extended Kricker invariant is a universal finite type invariant with respect to null Lagrangian-preserving surgeries.

Lescop defined in [Les11] an invariant of null-homologous knots in $\mathbb{Q}$–spheres and proved in [Les13] splitting formulas for this invariant with respect to null Lagrangian-preserving surgeries, similar to the ones proved in this paper for the Kricker invariant. Lescop conjectured in [Les13] that her invariant is equivalent to the Kricker invariant. The mentioned results of Garoufalidis, Kricker, Lescop and Rozansky give such an equivalence for knots in $\mathbb{Z}$–spheres with trivial Alexander polynomial and the results of the present paper allow to generalize this equivalence to null-homologous knots in $\mathbb{Q}$–spheres with trivial Alexander polynomial [Mou17, Theorem 2.11].

A similar situation arises in the study of finite type invariants of $\mathbb{Q}$–spheres with respect to Lagrangian-preserving surgeries. In this case, the Kontsevich–Kuperberg–Thurston invariant and the Le–Murakami–Ohtsuki invariant are both universal, up to degree 1 invariants deduced from the cardinality of the first homology group; this implies an equivalence result for these two invariants, see [Mou12]. For the KKT invariant, splitting formulas with respect to Lagrangian-preserving surgeries were proved by Lescop [Les04]; for the LMO invariant, similar formulas were proved by Massuyeau [Mas15]. Massuyeau’s proof of his splitting formulas is based on an extension of the LMO invariant of $\mathbb{Q}$–spheres to a functor defined on a category of Lagrangian cobordisms that he constructed with Cheptea and Habiro [CHM08].

In this paper, we extend the LMO functorial invariant of Cheptea–Habiro–Massuyeau to a category of Lagrangian cobordisms with paths, inserting the Kricker’s idea in the construction. We obtain a functorial invariant from which the Kricker invariant of null-homologous knots in $\mathbb{Q}$–spheres is recovered. Following Massuyeau, we use the functoriality to obtain splitting formulas for our invariant and, as a consequence, for the Kricker invariant.

Notations and conventions.

For $\mathbb{K}=\mathbb{Z},\mathbb{Q}$, a $\mathbb{K}$–sphere, (resp. a $\mathbb{K}$–cube) is a 3–manifold, compact and oriented, which has the same homology with coefficients in $\mathbb{K}$ as the standard 3–sphere (resp. 3–cube). The boundary of an oriented manifold with boundary is oriented with the “outward normal first” convention.

Acknowledgments.

I am supported by a Postdoctoral Fellowship of the Japan Society for the Promotion of Science. I am grateful to Tomotada Ohtsuki and the Research Institute for Mathematical Sciences for their support. I also wish to thank Gwénaël Massuyeau for interesting exchanges. Finally, I thank the referee whose useful comments helped to improve the paper.

1.2 Statement of the main result

We first give the definitions we need to state our main result.

Null LP–surgeries.

For $g\in\mathbb{N}$, a genus $g$ rational homology handlebody ($\mathbb{Q}$–handlebody) is a 3–manifold which is compact, oriented, and which has the same homology with rational coefficients as the standard genus $g$ handlebody. Such a $\mathbb{Q}$–handlebody is connected, and its boundary is necessarily homeomorphic to the standard genus $g$ surface.

The Lagrangian $\mathcal{L}_{C}$ of a $\mathbb{Q}$–handlebody $C$ is the kernel of the map $i_{*}:H_{1}(\partial C;\mathbb{Q})\to H_{1}(C;\mathbb{Q})$ induced by the inclusion. The Lagrangian of a $\mathbb{Q}$–handlebody $C$ is indeed a Lagrangian subspace of $H_{1}(\partial C;\mathbb{Q})$ with respect to the intersection form. A Lagrangian-preserving pair, or LP–pair, is a pair $\mathbf{C}=\left(\frac{C^{\prime}}{C}\right)$ of $\mathbb{Q}$–handlebodies equipped with a homeomorphism $h:\partial C\smash{\mathop{\hbox to31.29802pt{ \rightarrowfill\ }}\limits^{\scriptstyle{\cong}}}\partial C^{\prime}$ such that $h_{*}(\mathcal{L}_{C})=\mathcal{L}_{C^{\prime}}$.

Given a 3–manifold $M$, a Lagrangian-preserving surgery, or LP–surgery, on $M$ is a family $\mathbf{C}=(\mathbf{C}_{1},\dots,\mathbf{C}_{n})$ of LP–pairs such that the $C_{i}$ are embedded in $M$ and disjoint. The manifold obtained from $M$ by LP–surgery on $\mathbf{C}$ is defined as

[TABLE]

Let $M$ be a 3–manifold such that $H_{1}(M;\mathbb{Q})=0$ and let $K$ be a disjoint union of knots or paths properly embedded in $M$. A $\mathbb{Q}$–handlebody null in $M\setminus K$ is a $\mathbb{Q}$–handlebody $C\subset M\setminus K$ such that the map $i_{*}:H_{1}(C;\mathbb{Q})\to H_{1}(M\setminus K;\mathbb{Q})$ induced by the inclusion has a trivial image. A null LP–surgery on $(M,K)$ is an LP–surgery $\mathbf{C}=(\mathbf{C}_{1},\dots,\mathbf{C}_{n})$ on $M\setminus K$ such that each $C_{i}$ is null in $M\setminus K$. The pair obtained by surgery is denoted by $(M,K)(\mathbf{C})$.

The tensor $\mu(\mathbf{C})$.

Given an LP–pair $\mathbf{C}=\left(\frac{C^{\prime}}{C}\right)$, define the associated total manifold $\mathcal{C}=(-C)\cup C^{\prime}$ and define

[TABLE]

by associating with a triple of cohomology classes the evaluation of their triple cup product on the fundamental class of $\mathcal{C}$. For a family $\mathbf{C}=(\mathbf{C}_{1},\dots,\mathbf{C}_{n})$ of LP–pairs, let $\mathcal{C}=\mathcal{C}_{1}\sqcup\dots\sqcup\mathcal{C}_{n}$ and set:

[TABLE]

The natural identification $H_{1}(\mathcal{C};\mathbb{Q})\cong\oplus_{i=1}^{n}H_{1}(\mathcal{C}_{i};\mathbb{Q})$ allows to see $\otimes_{i=1}^{n}\Lambda^{3}H_{1}(\mathcal{C}_{i};\mathbb{Q})$ as a subspace of $\left(\Lambda^{3}H_{1}(\mathcal{C};\mathbb{Q})\right)^{\otimes n}$. This subspace injects to $S^{n}\Lambda^{3}H_{1}(\mathcal{C};\mathbb{Q})$ via the canonical surjection $\left(\Lambda^{3}H_{1}(\mathcal{C};\mathbb{Q})\right)^{\otimes n}\twoheadrightarrow S^{n}\Lambda^{3}H_{1}(\mathcal{C};\mathbb{Q})$. Hence we can view $\mu(\mathbf{C})$ as an element of $S^{n}\Lambda^{3}H_{1}(\mathcal{C};\mathbb{Q})$.

The bilinear form $\ell_{(S,\kappa)}(\mathbf{C})$.

Let $(S,\kappa)$ be a $\mathbb{Q}$SK–pair, i.e. a pair made of a $\mathbb{Q}$–sphere $S$ and a null-homologous knot $\kappa\subset S$. Let $\mathbf{C}=(\mathbf{C}_{1},\dots,\mathbf{C}_{n})$ be a null LP–surgery on $(S,\kappa)$. Let $\mathcal{C}=\mathcal{C}_{1}\sqcup\dots\sqcup\mathcal{C}_{n}$ be the disjoint union of the associated total manifolds. Fix a lift $\tilde{C}_{i}$ of each $C_{i}$ in $\tilde{E}$. We will define a hermitian form:

[TABLE]

i.e. a $\mathbb{Q}$–bilinear form such that reversing the order of the arguments changes $t$ to $t^{-1}$. Let $a\in H_{1}(\mathcal{C}_{i};\mathbb{Q})$ and $b\in H_{1}(\mathcal{C}_{j};\mathbb{Q})$ be homology classes that can be represented by simple closed curves $\alpha\subset\partial C_{i}$ and $\beta\subset\partial C_{j}$, disjoint if $i=j$. Note that such homology classes generate $H_{1}(\mathcal{C};\mathbb{Q})$ over $\mathbb{Q}$. Let $\tilde{\alpha}$ and $\tilde{\beta}$ be the copies of $\alpha$ and $\beta$ in $\tilde{C}_{i}$ and $\tilde{C}_{j}$. Set:

[TABLE]

where $\textrm{{lk}}_{e}(\cdot,\cdot)$ stands for the equivariant linking number (see for instance [Mou17, Section 2.1] for a definition). We get a well-defined hermitian form $\ell_{(S,\kappa)}(\mathbf{C})$ associated with a choice of lifts of the $C_{i}$’s. We will keep this choice implicit; the statement of Theorem 1.1 is valid for any such choice.

Diagrammatic representations.

Let $V$ be a rational vector space. A $V$–colored Jacobi diagram is a unitrivalent graph whose trivalent vertices are oriented and whose univalent vertices are labelled by $V$, where an orientation of a trivalent vertex is a cyclic order of the three edges that meet at this vertex —fixed as

in the pictures. Set:

[TABLE]

where the relations are depicted in Figure 1.

A symmetric tensor in $S^{n}\Lambda^{3}V$ can be represented by a Jacobi diagram via the following embedding.

[TABLE]

Now define a $\mathbb{Q}(t)$–beaded Jacobi diagram as a trivalent graph whose vertices are oriented and whose edges are oriented and labelled by $\mathbb{Q}(t)$. Set:

[TABLE]

where the relations are depicted in Figures 1 and 2,

with the IHX relation defined with the central edge labelled by 1. Define the i–degree, or internal degree, of any Jacobi diagram as its number of trivalent vertices. Given a hermitian form $\ell:V\times V\to\mathbb{Q}(t)$, one can glue with $\ell$ some legs of a $V$–colored Jacobi diagram as depicted in Figure 3.

If $n$ is even, one can pairwise glue all legs of a $V$–colored Jacobi diagram of i–degree $n$ in order to get an element of $\widetilde{\mathcal{A}}_{\mathbb{Q}(t)}(\varnothing)$. This latter space is the target space of the Kricker invariant of $\mathbb{Q}$SK–pairs.

We can now state our main result, about the Kricker invariant $\tilde{Z}$, proved in Section 6. Note that null LP–surgeries define a move on the set of $\mathbb{Q}$SK–pairs.

Theorem 1.1**.**

Let $(S,\kappa)$ be a $\mathbb{Q}$SK–pair. Let $\mathbf{C}=(\mathbf{C}_{1},\dots,\mathbf{C}_{n})$ be a null LP–surgery on $(S,\kappa)$. Then:

[TABLE]

where $\equiv_{n}$ means “equal up to terms of i–degree at least $n+1$”.

Example.

Let $(S^{3},\mathcal{O})$ be the $\mathbb{Q}$SK–pair defined by the trivial knot $\mathcal{O}$ in the standard $3$–sphere. Let $C_{1}$ and $C_{2}$ be regular neighborhoods of the graphs $\Gamma_{1}$ and $\Gamma_{2}$ drawn in Figure 4.

One can define an LP–surgery $\mathbf{C}=(\mathbf{C}_{1},\mathbf{C}_{2})$ by associating with each $\Gamma_{i}$ a Borromean surgery, see for instance [Mou17, Section 2.2]. The associated tensor is given by $\mu(\mathbf{C}_{1})=\zeta_{1}\wedge\zeta_{2}\wedge\zeta_{3}$ and $\mu(\mathbf{C}_{2})=\xi_{1}\wedge\xi_{2}\wedge\xi_{3}$. There are fifteen ways to glue all legs of

[TABLE]

with $\frac{1}{2}\ell_{(S^{3},\mathcal{O})}(\mathbf{C})$; all associated diagrams but one are trivial by the relation LE since they have a trivially labelled edge. Now $\textrm{{lk}}_{e}(\zeta_{1},\xi_{1})=1$, $\textrm{{lk}}_{e}(\zeta_{2},\xi_{2})=1$ and $\textrm{{lk}}_{e}(\zeta_{3},\xi_{3})=t^{-1}$, so that we finally get:

[TABLE]

where the vanishing of two terms in the left hand side is due to [GGP01, Lemma 2.2].

1.3 Strategy

In this Subsection, we give a rough overview of the strategy developed to prove Theorem 1.1.

The main object of this article is the construction of a functorial LMO invariant defined on a category of Lagrangian cobordisms with paths. The morphisms of this category are cobordisms between compact surfaces with one boundary component, satisfying a Lagrangian-preserving condition, with finitely many disjoint paths with fixed extremities which we think of as knots with a fixed part on the boundary. This category is equivalent to a category of bottom-top tangles in $\mathbb{Q}$–cubes, whose top part has a trivial linking matrix, with paths with fixed extremities. These bottom-top tangles can be viewed as morphisms in a category of (general) tangles with paths in $\mathbb{Q}$–cubes, with an important difference in the composition law. Now a tangle with paths in a $\mathbb{Q}$–cube can be expressed as the result of a surgery on a link in a tangle with trivial paths —segment lines— in $[-1,1]^{3}$. To sum up, with a Lagrangian cobordism with paths, we associate a tangle with disks —whose boundaries define the paths— in $[-1,1]^{3}$ with a surgery link. This is represented in the first line of the scheme in Figure 5. We initiate the construction of the invariant at the “tangle with disks” level.

On the above mentioned categories, we define functorial invariants valued in categories of Jacobi diagrams with beads, i.e. unitrivalent graphs whose univalent vertices are labelled by some finite set or embedded in some 1–manifold —the skeleton— and whose edges are labelled (beaded) by powers of $t$, polynomials in $\mathbb{Q}[t^{\pm 1}]$ or rational functions in $\mathbb{Q}(t)$. At the first step, we define a functor $Z^{\bullet}$ on the category of tangles with disks by applying the Kontsevich integral and adding a bead $t^{\pm 1}$ on the skeleton when the corresponding component meets a disk of the tangle. At a second step, we apply the invariant $Z^{\bullet}$ to surgery presentations of tangles with paths in $\mathbb{Q}$–cubes. We use the formal Gaussian integration methods introduced by Bar-Natan, Garoufalidis, Rozansky and Thurston in [BNGRT02a, BNGRT02b] and adapted to the beaded setting in [Kri00, GK04]. We get a functor $Z$ on the category of tangles with paths in $\mathbb{Q}$–cubes. At the last step, given a Lagrangian cobordism with paths, we apply $Z$ to the associated bottom-top tangles with paths and normalize it following [CHM08] to obtain a functor $\tilde{Z}$ on the category of Lagrangian cobordisms with paths. Functoriality allows to prove splitting formulas for this invariant with respect to null Lagrangian-preserving surgeries.

Given a Lagrangian cobordism with one path between genus 0 surfaces, i.e. a $\mathbb{Q}$–cube with one path, one can glue a 3–ball to the boundary to get a $\mathbb{Q}$–sphere with a knot. In this way, the functor $\tilde{Z}$ provides an invariant of $\mathbb{Q}$SK–pairs which coincides with the Kricker invariant for knots in $\mathbb{Z}$–spheres. Splitting formulas for this invariant are deduced from the splitting formulas for the functor $\tilde{Z}$.

Plan of the paper.

We define the domain categories of cobordisms and tangles in Section 2. In Section 3, we define the target categories of Jacobi diagrams and gives the tools of formal Gaussian integration. Section 4 is devoted to the introduction of winding matrices, that will play the role of the linking matrices in the Cheptea–Habiro–Massuyeau construction of a functorial LMO invariant. The functors $Z^{\bullet}$, $Z$ and $\tilde{Z}$ are constructed in Section 5. At the end of this section, from the functor $\tilde{Z}$, we deduce our version of the Kricker invariant for $\mathbb{Q}$SK–pairs; the behaviour of this invariant with respect to connected sum is stated. Finally, the splitting formulas are given in Section 6.

2 Domain categories: cobordisms and tangles

2.1 Cobordisms with paths

Given $g\in\mathbb{N}$, we fix a model surface $F_{g}$, compact, connected, oriented, of genus $g$, with one boundary component represented in Figure 6.

It is equipped with a fixed base point $*$ and a fixed basis $(\alpha_{1},\dots,\alpha_{g},\beta_{1},\dots,\beta_{g})$ of $\pi_{1}(F_{g},*)$. Denote by $C^{g^{+}}_{g^{-}}$ the cube $[-1,1]^{3}$ with $g^{+}$ handles on the top boundary and $g^{-}$ tunnels in the bottom boundary. We have canonical embeddings $F_{g^{+}}\hookrightarrow\partial C^{g^{+}}_{g^{-}}$ and $F_{g^{-}}\hookrightarrow\partial C^{g^{+}}_{g^{-}}$. For $k\geq 0$ and $1\leq i\leq k$, set $h_{i}(k)=\frac{2i}{k+1}-1$. A cobordism with paths from $F_{g^{+}}$ to $F_{g^{-}}$ is an equivalence class of triples $(M,K,m)$ where:

• •

$M$ is a compact, connected, oriented 3–manifold,

• •

$m:\partial C^{g^{+}}_{g^{-}}\smash{\mathop{\hbox to31.29802pt{ \rightarrowfill\ }}\limits^{\scriptstyle{\cong}}}\partial M$ is an orientation-preserving homeomorphism,

• •

$K=\sqcup_{i=1}^{k}K_{i}\subset M$ is a union of $k\geq 0$ oriented paths $K_{i}$ from $m(0,1,h_{i}(k))$ to $m(0,-1,h_{i}(k))$,

• •

$\hat{K}=\sqcup_{i=1}^{k}\hat{K}_{i}$ is an oriented boundary link, i.e. the $\hat{K}_{i}$ bound disjoint compact oriented surfaces in $M$, where $\hat{K}_{i}$ is the knot defined as the union of $K_{i}$ with the line segments $[(0,-1,h_{i}(k)),(1,-1,h_{i}(k))]$, $[(1,-1,h_{i}(k)),(1,1,h_{i}(k))]$ and $[(1,1,h_{i}(k)),(0,1,h_{i}(k))]$.

Two such triples are equivalent if they are related by an orientation-preserving homeomorphism which respects the boundary parametrizations and identifies the paths. We get embeddings $m_{+}:F_{g^{+}}\hookrightarrow\partial M$ and $m_{-}:F_{g^{-}}\hookrightarrow\partial M$.

Define a category $\widetilde{\mathcal{C}ob}$ of cobordisms with paths whose objects are non-negative integers and whose set of morphisms $\widetilde{\mathcal{C}ob}(g^{+},g^{-})$ is the set of cobordisms with paths from $F_{g^{+}}$ to $F_{g^{-}}$. The composition of a cobordism $(M,K,m)$ from $F_{g}$ to $F_{f}$ with a cobordism $(N,J,n)$ from $F_{h}$ to $F_{g}$ is given by gluing $N$ on the top of $M$, identifying $m_{+}(M)$ with $n_{-}(N)$ and reparametrizing the new manifold. The identity of $g\in\mathbb{N}$ is the cobordism $F_{g}\times[-1,1]$ with natural boundary parametrization and no path.

Forgetting the datum of the paths in the cobordisms, one gets the category $\mathcal{C}ob$ described in [CHM08], which we view as the subcategory of $\widetilde{\mathcal{C}ob}$ of cobordisms with no path. For a cobordism $(M,m)$ and a cobordism with paths $(N,J,n)$, define the tensor product $(M,m)\otimes(N,J,n)$ by horizontal juxtaposition in the $x$ direction.

We now define the subcategory of $\widetilde{\mathcal{C}ob}$ of Lagrangian cobordisms with paths. Set $A_{g}=\ker(\mathrm{incl}_{*}:H_{1}(F_{g};\mathbb{Q})\to H_{1}(C^{g}_{0};\mathbb{Q}))$ and $B_{g}=\ker(\mathrm{incl}_{*}:H_{1}(F_{g};\mathbb{Q})\to H_{1}(C^{0}_{g};\mathbb{Q}))$. These are Lagrangian subspaces of $H_{1}(F_{g};\mathbb{Q})$ with respect to the intersection form and $A_{g}$ (resp. $B_{g}$) is generated by the homology classes of the curves $\alpha_{i}$ (resp. $\beta_{i}$). A cobordism with paths $(M,K,m)$ from $F_{g^{+}}$ to $F_{g^{-}}$ is Lagrangian(-preserving) if the following conditions are satisfied:

• •

$H_{1}(M;\mathbb{Q})=(m_{-})_{*}(A_{g^{-}})\oplus(m_{+})_{*}(B_{g^{+}})$,

• •

$(m_{+})_{*}(A_{g^{+}})\subset(m_{-})_{*}(A_{g^{-}})$ as subspaces of $H_{1}(M;\mathbb{Q})$.

The Lagrangian property is preserved by composition, and we denote by $\widetilde{\mathcal{LC}ob}$ the subcategory of $\widetilde{\mathcal{C}ob}$ of Lagrangian cobordisms with paths. The subcategory of Lagrangian cobordisms with no path is the category $\mathcal{LC}ob$ —denoted by $\mathbb{Q}\mathcal{LC}ob$ in [Mas15].

Define categories $\mathcal{C}ob_{q}$, $\widetilde{\mathcal{C}ob}_{q}$, $\mathcal{LC}ob_{q}$ and $\widetilde{\mathcal{LC}ob}_{q}$ of $q$–cobordisms with objects the non-commutative words in the single letter $\bullet$ and with set of morphisms from a word on $g^{+}$ letters to a word on $g^{-}$ letters the set of morphisms from $g^{+}$ to $g^{-}$ in $\mathcal{C}ob$, $\widetilde{\mathcal{C}ob}$, $\mathcal{LC}ob$ and $\widetilde{\mathcal{LC}ob}$ respectively.

Let $(M,K,m)$ be a Lagrangian cobordism with paths from $g^{+}$ to $g^{-}$. Since the space $H_{1}(M;\mathbb{Q})$ is non-trivial in general, we have to adapt the definition of null LP–surgeries given in the introduction. Let $N(K)$ be a tubular neighborhood of $K$ in $M$ and set $E=M\setminus\textrm{Int}(N(K))$. A standard homological computation gives $H_{2}(M;\mathbb{Q})=0$, so that the exact sequence in homology associated with the pair $(M,E)$ provides the following short exact sequence:

[TABLE]

The image of $H_{2}(N(K),N(K)\cap E;\mathbb{Q})$ in $H_{1}(E;\mathbb{Q})$ is a subspace $H_{K}\cong\mathbb{Q}^{k}$ generated by meridians of the components of $K$, where $k$ is the number of these components. Now, the parametrizations $m_{\pm}$ of the top and bottom boundaries of $M$ can be decomposed into injective maps as follows.

E$$F_{g^{\pm}}$$M$$m_{\pm}$$e_{\pm}

Hence we have a canonical decomposition of $H_{1}(E;\mathbb{Q})$ as

[TABLE]

where $(e_{-})_{*}(A_{g^{-}})\oplus(e_{+})_{*}(B_{g^{+}})\cong H_{1}(M;\mathbb{Q})$ via the inclusion map. We say that a $\mathbb{Q}$–handlebody $C\subset M\setminus K$ is null with respect to $K$ if the composed map

[TABLE]

has a trivial image, where the second map is the projection on the first factor in the above decomposition of $H_{1}(M\setminus K;\mathbb{Q})\cong H_{1}(E;\mathbb{Q})$. A null LP–surgery on $(M,K)$ is an LP–surgery $\mathbf{C}=(\mathbf{C}_{1},\dots,\mathbf{C}_{n})$ on $M\setminus K$ such that each $C_{i}$ is null with respect to $K$.

2.2 Bottom-top tangles with paths

Let us define the category of bottom-top tangles with paths. For a positive integer $g\geq 0$, let $(p_{1},q_{1}),\dots,(p_{g},q_{g})$ be $g$ pairs of points uniformly distributed on $[-1,1]\times\{0\}\subset[-1,1]^{2}\cong F_{0}$ as represented in Figure 8.

A bottom-top tangle with paths of type $(g^{+},g^{-})$ is an equivalence class of triples $(B,K,\gamma)$ where:

• •

$(B,K)=(B,K,b)$ is a cobordism with paths form $F_{0}$ to $F_{0}$,

• •

$\gamma=(\gamma^{+},\gamma^{-})$ is a framed oriented tangle in $B$ with $g^{+}$ components $\gamma_{i}^{+}$ from $b(\{p_{i}\}\times\{1\})$ to $b(\{q_{i}\}\times\{1\})$ and $g^{-}$ components $\gamma_{i}^{-}$ from $b(\{q_{i}\}\times\{-1\})$ to $b(\{p_{i}\}\times\{-1\})$,

• •

$\hat{K}$ is a boundary link in $B\setminus\gamma$.

Two such triples $(B,K,\gamma)$ and $(B^{\prime},K^{\prime},\gamma^{\prime})$ are equivalent if $(B,K)$ and $(B^{\prime},K^{\prime})$ are related by an equivalence which identifies $\gamma$ and $\gamma^{\prime}$.

In order to define the composition, we need the bottom-top tangle $([-1,1]^{3},\varnothing,T_{g})$ represented in Figure 9.

The composition of a bottom-top tangle $(B,K,\gamma)$ of type $(g,f)$ with a bottom-top tangle $(C,J,\upsilon)$ of type $(h,g)$ is given by first making the composition $(B,K)\circ([-1,1]^{3},\varnothing)\circ(C,J)$ in the category $\widetilde{\mathcal{C}ob}$ and then perfoming the surgery on the $2g$ components link $\gamma^{+}\cup T_{g}\cup\upsilon^{-}$. We get a category ${}_{b}^{t}\widetilde{\mathcal{T}}$ whose objects are non-negative integers and whose set of morphisms ${}_{b}^{t}\widetilde{\mathcal{T}}(g^{+},g^{-})$ is the set of bottom-top tangles with paths of type $(g^{+},g^{-})$. The identity of $g\in\mathbb{N}$ is the bottom-top tangle in $[-1,1]^{3}$ with no path represented in Figure 10.

Forgetting the datum of the paths, one gets the category ${}_{b}^{t}\mathcal{T}$ of bottom-top tangles introduced in [CHM08], that we view as the subcategory of ${}_{b}^{t}\widetilde{\mathcal{T}}$ of bottom-top tangles with no path. For a bottom-top tangle $(B,\gamma)$ and a bottom-top tangle with paths $(C,J,\upsilon)$, define the tensor product $(B,\gamma)\otimes(C,J,\upsilon)$ by horizontal juxtaposition in the $x$ direction. Define categories ${}_{b}^{t}\mathcal{T}_{q}$ and ${}_{b}^{t}\widetilde{\mathcal{T}}_{q}$ of bottom-top $q$–tangles with objects the non-commutative words in the single letter $\bullet$.

The following result is a direct adaptation of [CHM08, Theorem 2.10] which gives an isomorphism $D:{}_{b}^{t}\mathcal{T}\to\mathcal{C}ob$. The map $D$ is defined by digging tunnels around the components of the tangle.

Proposition 2.1**.**

There is an isomorphism $D:{}_{b}^{t}\widetilde{\mathcal{T}}\to\widetilde{\mathcal{C}ob}$ which identifies ${}_{b}^{t}\mathcal{T}$ with $\mathcal{C}ob$ and preserves the tensor product on ${}_{b}^{t}\mathcal{T}\otimes{}_{b}^{t}\widetilde{\mathcal{T}}$.

Let $(B,K,\gamma)$ be a bottom-top tangle with paths in a $\mathbb{Q}$–cube. Let $\bar{\gamma}$ be the link obtained by closing the components of $\gamma$ with the line segments $[(p_{i},\pm 1),(q_{i},\pm 1)]$. Define the linking matrix $\textrm{{Lk}}(\gamma)$ of $\gamma$ in $B$ with the linkings of the components of $\bar{\gamma}$. The characterization of the bottom-top tangles sent onto Lagrangian cobordisms by $D$ given in [CHM08, Lemma 2.12] directly generalizes to:

Lemma 2.2**.**

Given a bottom-top tangle with paths $(B,K,\gamma)$, the cobordism with paths $D(B,K,\gamma)$ is Lagrangian if and only if $B$ is a $\mathbb{Q}$–cube and $\textrm{{Lk}}(\gamma^{+})$ is trivial.

2.3 Tangles with paths and tangles with disks

Given a cobordism $(B,b)$ from $F_{0}$ to $F_{0}$, a tangle $\gamma$ in $B$ is an isotopy (rel. $\partial B$) class of framed oriented tangles whose boundary points lie on the top and bottom surfaces and are uniformly distributed along the line segments $[-1,1]\times\{0\}\times\{1\}$ and $[-1,1]\times\{0\}\times\{-1\}$ in $\partial B=b(\partial[-1,1]^{3})$. Associate with each boundary point of $\gamma$ the sign $+$ if $\gamma$ is oriented downwards at that point and the sign $-$ otherwise. This provides two words in the letters $+$ and $-$, one for the top surface and the other for the bottom surface. Lifting these two words into non-associative words $w_{t}(\gamma)$ and $w_{b}(\gamma)$ in the letters $(+,-)$, one gets a $q$–tangle. A $q$–tangle $\gamma$ in a cobordism with paths $(B,K,b)$ defines a $q$–tangle with paths $(B,K,\gamma)$ if $\hat{K}$ is a boundary link in $B\setminus\gamma$.

Define two categories $\mathcal{T}_{q}\mathcal{C}ub$ and $\widetilde{\mathcal{T}}_{q}\mathcal{C}ub$ with objects the non-associative words in the letters $(+,-)$ and morphisms the $q$–tangles in $\mathbb{Q}$–cubes for $\mathcal{T}_{q}\mathcal{C}ub$ and the $q$–tangles with paths in $\mathbb{Q}$–cubes for $\widetilde{\mathcal{T}}_{q}\mathcal{C}ub$, up to orientation-preserving homeomorphism respecting the boundary parametrization. Composition is given by vertical juxtaposition. Given a morphism $(C,\upsilon)$ in $\mathcal{T}_{q}\mathcal{C}ub$ and a morphism $(B,K,\gamma)$ in $\widetilde{\mathcal{T}}_{q}\mathcal{C}ub$, define the tensor product $(C,\upsilon)\otimes(B,K,\gamma)$ by horizontal juxtaposition in the $x$ direction.

Lemma 2.3**.**

Let $(B,K,\gamma)$ be a $q$–tangle with paths in a $\mathbb{Q}$–cube. There exist a $q$–tangle with paths $([-1,1]^{3},\Xi,\eta)$ and a framed link $L\subset[-1,1]^{3}\setminus(\Xi\cup\eta)$, with $\Xi$ a union of line segments and $L$ null-homotopic in $[-1,1]^{3}\setminus\Xi$, such that $(B,K,\gamma)$ is obtained from $([-1,1]^{3},\Xi,\eta)$ by surgery on $L$. Moreover, two such surgery links are related by the following Kirby moves: the blow-up/blow-down move KI which adds or removes a split trivial component with framing $\pm 1$ unknotted with $\eta$, and the handleslide move KII which adds a surgery component to another surgery component or to a component of the tangle $\eta$ (see Figure 11).

Proof.

Let $\Sigma$ be a Seifert surface of $\hat{K}$ which is the disjoint union of Seifert surfaces of the $\hat{K}_{i}$. Choose $\Sigma$ disjoint from $\gamma$. Take a link $J\subset B$ such that surgery on $J$ gives $[-1,1]^{3}$. Performing isotopies on $J$ if necessary, we can assume that $J$ does not meet $\Sigma$. The handles of (the image in $[-1,1]^{3}$ of) $\Sigma$ can be unlinked by adding surgery components as shown in Figure 12.

In this way, $\hat{K}$ can be turned into a trivial link. This provides a surgery link in $B\setminus(K\cup\gamma)$, disjoint from $\Sigma$, such that surgery on this link changes $(B,K,\gamma)$ to a $q$–tangle with paths $([-1,1]^{3},\Xi,\eta)$ as required. Let $L$ be an inverse surgery link. It is null-homotopic in $[-1,1]^{3}\setminus\Xi$ since it is disjoint from $\Sigma$.

For the last assertion, apply [HW14, Theorem 3.1] in $[-1,1]^{3}\setminus\Xi$. Note that a split trivial component with framing $\pm 1$ can always be unknotted from $\eta$ using the KII move. ∎

A family $(([-1,1]^{3},\Xi,\eta),L)$ satisfying the conditions of the lemma with $L$ oriented is a surgery presentation of $(B,K,\gamma)$. When $\gamma$ (and thus $\eta$) is a bottom-top tangle, the components of $\eta$ can be closed by line segments in the top and bottom surfaces. The obtained curves are null-homotopic in $[-1,1]^{3}\setminus\Xi$ since $\hat{\Xi}$ is a boundary link in $[-1,1]^{3}\setminus\eta$.

The notion of a $q$–tangle in $[-1,1]^{3}$ with trivial paths, i.e. line segments, is equivalent to the following one. A $q$–tangle with disks is an equivalence class of pairs $(\gamma,k)$, where $\gamma$ is a $q$–tangle in $[-1,1]^{3}$, $k$ is a non-negative integer understood as the datum of $k$ disks $d_{i}=[0,1]\times[-1,1]\times\{h_{i}(k)\}$ and the link $\sqcup_{i=1}^{k}\partial d_{i}$ associated with the paths $d_{i}^{\partial}=\{0\}\times[-1,1]\times\{h_{i}(k)\}$ is a boundary link in $[-1,1]^{3}\setminus\gamma$. An example of such a tangle with disks is drawn in Figure 13, projected in the $y$ direction. Equivalence of such pairs is defined as isotopy relative to $(\partial[-1,1]^{3})\cup(\cup_{i=1}^{k}d_{i}^{\partial})$. Define two categories $\mathcal{T}_{q}$ and $\widetilde{\mathcal{T}}_{q}$ with objects the non-associative words in the letters $(+,-)$ and morphisms the $q$–tangles for $\mathcal{T}_{q}$ and the $q$–tangles with disks for $\widetilde{\mathcal{T}}_{q}$. Composition is given by vertical juxtaposition. Given a $q$–tangle $\gamma$ and a $q$–tangle with disks $(\upsilon,k)$, define the tensor product $\gamma\otimes(\upsilon,k)$ in $\widetilde{\mathcal{T}}_{q}((w_{t}(\gamma))(w_{t}(\upsilon)),(w_{b}(\gamma))(w_{b}(\upsilon)))$ by horizontal juxtaposition in the $x$ direction. Define similarly two categories $\mathcal{T}$ and $\widetilde{\mathcal{T}}$ of tangles and tangles with disks in $[-1,1]^{3}$ with objects the associative words in the letters $(+,-)$.

3 Target categories: Jacobi diagrams with beads

3.1 Diagram spaces

For a compact oriented 1–manifold $X$ and a finite set $C$, a Jacobi diagram on $(X,C)$ is a unitrivalent graph whose trivalent vertices are oriented and whose univalent vertices are embedded in $X$ or labelled by $C$, where an orientation of a trivalent vertex is a cyclic order of the three edges that meet at this vertex —fixed as

in the pictures. The manifold $X$ is called the skeleton of the diagram. Next, let $R$ be the ring $\mathbb{Q}[t^{\pm 1}]$ or $\mathbb{Q}(t)$. An $R$–beaded Jacobi diagram on $(X,C)$ is a Jacobi diagram on $(X,C)$ whose graph edges are oriented and labelled by $R$. Last, an $R$–winding Jacobi diagram on $(X,C)$ is an $R$–beaded Jacobi diagram on $(X,C)$ whose skeleton is viewed as a union of edges —defined by the embedded vertices— that are labelled by powers of $t$, with the condition that the product of the labels on each component of $X$ is 1. As defined in the introduction, the i–degree of a trivalent diagram is its number of trivalent vertices. Set:

[TABLE]

[TABLE]

[TABLE]

with the relations in Figures 1, 2 and 14, where the IHX relation for beaded and winding diagrams is defined with the central edge labelled by 1.

In the STU relation, the edges corresponding to each other have the same orientation and label. In the pictures, the skeleton is represented with full lines and the graph with dashed lines. We indeed consider the i–degree completion of these vector spaces, keeping the same notation.

Remark*.*

For diagrams in $\widetilde{\mathcal{A}}^{\textrm{{w}}}_{R}(X,*_{C})$, the condition on the labels on the skeleton implies that all labels can be pushed off each component of the skeleton using the Hol${}_{\textrm{w}}$ relation. When the component is an interval, there is a unique way to do so. Hence, when $X$ contains only intervals, $\widetilde{\mathcal{A}}^{\textrm{{w}}}_{R}(X,*_{C})$ is isomorphic to $\widetilde{\mathcal{A}}_{R}(X,*_{C})$.

For a finite set $S$, denote by $\,\raisebox{-2.15277pt}{ \leavevmode\hbox to0.4pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{10.92093pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{10.92093pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,_{S}$ (resp. $\raisebox{-0.86108pt}{ \leavevmode\hbox to9.51pt{\vbox to9.51pt{\pgfpicture\makeatletter\hbox{\hskip 4.75253pt\lower-4.75253pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@curveto{4.55254pt}{2.51433pt}{2.51433pt}{4.55254pt}{0.0pt}{4.55254pt}\pgfsys@curveto{-2.51433pt}{4.55254pt}{-4.55254pt}{2.51433pt}{-4.55254pt}{0.0pt}\pgfsys@curveto{-4.55254pt}{-2.51433pt}{-2.51433pt}{-4.55254pt}{0.0pt}{-4.55254pt}\pgfsys@curveto{2.51433pt}{-4.55254pt}{4.55254pt}{-2.51433pt}{4.55254pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{4.55254pt}{0.85355pt}\pgfsys@lineto{4.55254pt}{0.67792pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{4.55254pt}{0.67792pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,_{S}$) the manifold made of $|S|$ intervals (resp. circles) indexed by the elements of $S$. In the following, $\bar{\mathcal{A}}$ stands for $\mathcal{A}$, $\widetilde{\mathcal{A}}_{R}$ or $\widetilde{\mathcal{A}}^{\textrm{{w}}}_{R}$. In [BN95, Theorem 8], Bar-Natan defines a formal PBW isomorphism:

[TABLE]

For a Jacobi diagram $D$, the image $\chi_{S}(D)$ is the average of all possible ways to attach the $s$–colored vertices of $D$ on the corresponding $s$–indexed interval in $\,\raisebox{-2.15277pt}{ \leavevmode\hbox to0.4pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{10.92093pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{10.92093pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,_{S}$ for each $s\in S$. The setting of [BN95] is not exactly the same, but the argument adapts directly. When $|S|=1$, closing the $S$–labelled component gives an isomorphism from $\bar{\mathcal{A}}(X\cup\,\raisebox{-2.15277pt}{ \leavevmode\hbox to0.4pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{10.92093pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{10.92093pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,_{S},*_{C})$ to $\bar{\mathcal{A}}(X\cup\raisebox{-0.86108pt}{ \leavevmode\hbox to9.51pt{\vbox to9.51pt{\pgfpicture\makeatletter\hbox{\hskip 4.75253pt\lower-4.75253pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@curveto{4.55254pt}{2.51433pt}{2.51433pt}{4.55254pt}{0.0pt}{4.55254pt}\pgfsys@curveto{-2.51433pt}{4.55254pt}{-4.55254pt}{2.51433pt}{-4.55254pt}{0.0pt}\pgfsys@curveto{-4.55254pt}{-2.51433pt}{-2.51433pt}{-4.55254pt}{0.0pt}{-4.55254pt}\pgfsys@curveto{2.51433pt}{-4.55254pt}{4.55254pt}{-2.51433pt}{4.55254pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{4.55254pt}{0.85355pt}\pgfsys@lineto{4.55254pt}{0.67792pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{4.55254pt}{0.67792pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,_{S},*_{C})$ [BN95, Lemma 3.1]. However, this isomorphism does not hold for $|S|>1$. To recover an isomorphism onto $\bar{\mathcal{A}}(X\cup\raisebox{-0.86108pt}{ \leavevmode\hbox to9.51pt{\vbox to9.51pt{\pgfpicture\makeatletter\hbox{\hskip 4.75253pt\lower-4.75253pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@curveto{4.55254pt}{2.51433pt}{2.51433pt}{4.55254pt}{0.0pt}{4.55254pt}\pgfsys@curveto{-2.51433pt}{4.55254pt}{-4.55254pt}{2.51433pt}{-4.55254pt}{0.0pt}\pgfsys@curveto{-4.55254pt}{-2.51433pt}{-2.51433pt}{-4.55254pt}{0.0pt}{-4.55254pt}\pgfsys@curveto{2.51433pt}{-4.55254pt}{4.55254pt}{-2.51433pt}{4.55254pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{4.55254pt}{0.85355pt}\pgfsys@lineto{4.55254pt}{0.67792pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{4.55254pt}{0.67792pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,_{S},*_{C})$, some “link relations” were introduced in [BNGRT02b, Section 5.2]. We recall these relations and introduce additional “winding relations”.

Given a (beaded, winding) Jacobi diagram $D$ on $(X,C\cup S)$ and a univalent vertex $*$ of $D$ labelled by $s\in S$, define the associated link relation as the vanishing of the sum of all diagrams obtained from $D$ by gluing the vertex $*$ on the edges adjacent to a univalent $s$–labelled vertex, as follows:

\scriptscriptstyle{\bullet}$$\scriptscriptstyle{\bullet}$$*$$s , see Figure 15 (we omit the orientation of the edges when it is not relevant thanks to the OR relation).

Given a winding Jacobi diagram $D$ on $(X,C\cup S)$, a label $s\in S$ and an integer $k$, the associated winding relation identifies $D$ with the diagram obtained from $D$ by pushing $t^{k}$ at each $s$–labelled vertex, i.e. by multiplying the label of each edge adjacent to a univalent $s$–labelled vertex by $t^{k}$ if the orientation of the edge goes backward the vertex and by $t^{-k}$ otherwise, see Figure 16.

Denote $\bar{\mathcal{A}}(X,*_{C},\textrm{\footnotesize\raisebox{-0.9ex}{\normalsize*}⃝}_{S})$ (resp. $\bar{\mathcal{A}}(X,*_{C},\textrm{\scriptsize\raisebox{0.2ex}{\textcircled{\raisebox{-1.3ex}{\normalsize*}}}⃝}_{S})$) the quotient of $\bar{\mathcal{A}}(X,*_{C\cup S})$ by all link relations (resp. all link and winding relations) on $S$–labelled vertices. Note that if $X$ contains no closed component, then the spaces $\widetilde{\mathcal{A}}^{\textrm{{w}}}_{R}(X,*_{C},\textrm{\scriptsize\raisebox{0.2ex}{\textcircled{\raisebox{-1.3ex}{\normalsize*}}}⃝}_{S})$ and $\widetilde{\mathcal{A}}_{R}(X,*_{C},\textrm{\scriptsize\raisebox{0.2ex}{\textcircled{\raisebox{-1.3ex}{\normalsize*}}}⃝}_{S})$ are isomorphic. When some of the sets $X$, $C$, $S$ are empty, we simply drop the corresponding notation, mentionning $\varnothing$ only when they are all empty.

Proposition 3.1**.**

The isomorphisms $\chi_{S}:\bar{\mathcal{A}}(X,*_{C\cup S})\smash{\mathop{\hbox to31.29802pt{ \rightarrowfill\ }}\limits^{\scriptstyle{\scriptstyle{\cong}}}}\bar{\mathcal{A}}(X\cup\,\raisebox{-2.15277pt}{ \leavevmode\hbox to0.4pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{10.92093pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{10.92093pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,_{S},*_{C})$ descend to isomorphisms:

[TABLE]

[TABLE]

[TABLE]

Proof.

In the case $\bar{\mathcal{A}}=\mathcal{A}\textrm{ or }\widetilde{\mathcal{A}}_{R}$, it is [BNGRT02b, Theorem 3]. We recall briefly their argument in order to add the consideration of the winding relations when $\bar{\mathcal{A}}=\widetilde{\mathcal{A}}^{\textrm{{w}}}_{R}$.

The fact that the images by $\chi_{S}$ of the link relations map to 0 in $\bar{\mathcal{A}}(X\cup\raisebox{-0.86108pt}{ \leavevmode\hbox to9.51pt{\vbox to9.51pt{\pgfpicture\makeatletter\hbox{\hskip 4.75253pt\lower-4.75253pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@curveto{4.55254pt}{2.51433pt}{2.51433pt}{4.55254pt}{0.0pt}{4.55254pt}\pgfsys@curveto{-2.51433pt}{4.55254pt}{-4.55254pt}{2.51433pt}{-4.55254pt}{0.0pt}\pgfsys@curveto{-4.55254pt}{-2.51433pt}{-2.51433pt}{-4.55254pt}{0.0pt}{-4.55254pt}\pgfsys@curveto{2.51433pt}{-4.55254pt}{4.55254pt}{-2.51433pt}{4.55254pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{4.55254pt}{0.85355pt}\pgfsys@lineto{4.55254pt}{0.67792pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{4.55254pt}{0.67792pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,_{S},*_{C})$ follows from the STU relation. For the winding relations, it follows from the Hol${}_{\textrm{w}}$ relation applied at each univalent vertex glued on the $s$–labelled component, where $s$ is the label involved in the relation. Now, take two diagrams in $\bar{\mathcal{A}}(X\cup\,\raisebox{-2.15277pt}{ \leavevmode\hbox to0.4pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{10.92093pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{10.92093pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,_{S},*_{C})$ that are identified when closing an $S$–labelled component. We have to consider the two situations depicted in Figures 17 and 18, where the gray zone represents a hidden part of the diagram.

Equalities are obtained by applying STU relations in the first case and Hol${}_{\textrm{w}}$ relations in the second case.

Application of $\chi^{-1}$ to the right members provides linear combinations of the same sum with the skeleton component dropped and possibly trees glued. Using the IHX relation in the first case (resp. the Hol relation in the second case), we obtain link relations (resp. winding relations). ∎

3.2 Product and coproduct

We first define a coproduct on the diagram spaces of the previous subsection. Given a (beaded, winding) Jacobi diagram $D$ on $(X,C)$, denote by $\dddot{D}$ its graph part, and by $\dddot{D}_{i}$, $i\in I$, the connected components of $\dddot{D}$. Set $D_{J}=D\setminus(\sqcup_{i\in I\setminus J}\dddot{D}_{i})$. In the winding case, multiply the labels of the concatenated edges of the skeleton. Define the coproduct of a diagram $D$ by

[TABLE]

Note that the different relations on Jacobi diagrams respect the coproduct. This provides a notion of group-like elements, i.e. elements $G$ such that $\Delta(G)=G\otimes G$.

Set $\bar{\mathcal{A}}=\mathcal{A}\textrm{ or }\widetilde{\mathcal{A}}_{R}$. We will define a Hopf algebra structure on $\bar{\mathcal{A}}(*_{C})$. Define the product of two diagrams as the disjoint union. The unit $\epsilon:\mathbb{Q}\to\bar{\mathcal{A}}(*_{C})$ is defined by $\epsilon(1)=\varnothing$ and the counit $\varepsilon:\bar{\mathcal{A}}(*_{C})\to\mathbb{Q}$ is given by $\varepsilon(D)=0$ if $D\neq\varnothing$ and $\varepsilon(\varnothing)=1$. The antipode is given by $D\mapsto(-1)^{|I|}D$. We finally have a structure of a graded Hopf algebra on $\bar{\mathcal{A}}(*_{C})$, where the grading is given by the i–degree. It is known that an element in a graded Hopf algebra is group-like if and only if it is the exponential of a primitive element, i.e. an element $G$ such that $\Delta(G)=1\otimes G+G\otimes 1$. Here, the primitive elements are the series of connected diagrams.

The isomorphisms $\chi$ of the previous subsection are not algebra morphisms, but they preserve the coproduct. For the spaces $\bar{\mathcal{A}}(*_{C})$ with $\bar{\mathcal{A}}=\mathcal{A}\textrm{ or }\widetilde{\mathcal{A}}_{R}$, we have an exponential map associated with the product. For general (beaded, winding) Jacobi diagrams, we will use the notation $\textrm{{exp}}_{\sqcup}$, namely exponential with respect to the disjoint union, for linear combination of diagrams with no univalent vertex embedded in the skeleton, where the disjoint union applies only to the graph part.

3.3 Formal Gaussian integration

This part aims at defining a formal Gaussian integration along $S$ on $\widetilde{\mathcal{A}}^{\textrm{{w}}}_{R}(X,*_{C\cup S})$.

Definition 3.2**.**

A (beaded, winding) Jacobi diagram on $(X,C\cup S)$ is substantial if it has no strut, i.e. no isolated dashed edge. It is $S$–substantial if it has no $S$–strut, i.e. no strut with both vertices labelled in $S$.

Given two (beaded, winding) Jacobi diagrams $D$ and $E$ on $(X,C\cup S)$, one of whose is $S$–substantial, define $\langle D,E\rangle_{S}$ as the sum of all diagrams obtained by gluing all $s$–colored vertices of $D$ with all $s$–colored vertices of $E$ for all $s\in S$ —if the numbers of $s$–colored vertices in $D$ and $E$ do not match for some $s\in S$, then $\langle D,E\rangle_{S}=0$. In the beaded and winding cases, we must precise the orientation and label of the created edges. Such an edge is the gluing of two or three edges in the initial diagrams. Fix arbitrarily the orientation of the new edge. Let $P(t)$ (resp. $Q(t)$) be the product of the labels of the initial edges whose orientation coincides (resp. does not coincide). Define the label of the new edge as $P(t)Q(t^{-1})$, see Figure 19.

We have the following immediate lemma.

Lemma 3.3**.**

If $D^{\prime}$ and $E^{\prime}$ are obtained from $D$ and $E$ by applying the same winding relation on $s$–labelled vertices for some $s\in S$, then $\langle D,E\rangle_{S}=\langle D^{\prime},E^{\prime}\rangle_{S}$.

This bracketting defines a $\mathbb{Q}$–bilinear operator on the diagram spaces $\bar{\mathcal{A}}(X,*_{C\cup S})$ for $\bar{\mathcal{A}}=\mathcal{A},\ \widetilde{\mathcal{A}}_{R},\textrm{ or }\widetilde{\mathcal{A}}^{\textrm{{w}}}_{R}$.

Theorem 3.4** (Jackson, Moffatt, Morales).**

Assume the $1$–manifold $X$ is a disjoint union of intervals. If $G$ and $H$ are group-like in $\bar{\mathcal{A}}(X,*_{C\cup S})$, then $\langle G,H\rangle_{S}$ is also group-like.

Proof.

When $X=\varnothing$, it is [JMM06, Theorem 2.4]. The case of a non-empty $X$ follows since the isomorphism $\chi_{\pi_{0}(X)}:\bar{\mathcal{A}}(*_{\pi_{0}(X)\cup C\cup S})\smash{\mathop{\hbox to31.29802pt{ \rightarrowfill\ }}\limits^{\scriptstyle{\scriptstyle{\cong}}}}\bar{\mathcal{A}}(X,*_{C\cup S})$ preserves the coproduct and the bracketting $\langle\cdot,\cdot\rangle_{S}$. ∎

Notation*.*

If $W=(W_{ij}(t))_{i,j\in S}$ is an $(S,S)$–matrix with coefficients in $\mathbb{Q}(t)$, we also denote $W=\sum_{i,j\in S}$

\scriptscriptstyle{i}$$\scriptscriptstyle{j}$$\scriptscriptstyle{W_{ij}(t)} .

Definition 3.5**.**

An element $G\in\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}[t^{\pm 1}]}(X,*_{C\cup S})$ is Gaussian if $G=\textrm{{exp}}_{\sqcup}(\frac{1}{2}W(t))\sqcup H$ where $W(t)$ is an $(S,S)$–matrix with coefficients in $\mathbb{Q}[t^{\pm 1}]$ and $H$ is $S$–substantial. If $\det(W(t))\neq 0$, $G$ is non-degenerate and we set:

[TABLE]

Lemma 3.6**.**

Let $G=\textrm{{exp}}_{\sqcup}(\frac{1}{2}W(t))\sqcup H$ be a non-degenerate Gaussian in $\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}[t^{\pm 1}]}(X,*_{C\cup S})$.

• •

If a non-degenerate Gaussian $\textrm{{exp}}_{\sqcup}(\frac{1}{2}W(t))\sqcup H^{\prime}$ is equal to $G$ in $\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}[t^{\pm 1}]}(X,*_{C},\textrm{\footnotesize\raisebox{-0.9ex}{\normalsize*}⃝}_{S})$, then $\int_{S}(\textrm{{exp}}_{\sqcup}(\frac{1}{2}W(t))\sqcup H^{\prime})=\int_{S}G$.

• •

If $G^{\prime}=\textrm{{exp}}_{\sqcup}(\frac{1}{2}W^{\prime}(t))\sqcup H^{\prime}$ is obtained from $G=\textrm{{exp}}_{\sqcup}(\frac{1}{2}W(t))\sqcup H$ by applying a winding relation, then $\int_{S}G^{\prime}=\int_{S}G$.

Proof.

The first point is essentially given by the proof of Bar-Natan and Lawrence [BNL04, Proposition 2.2] in the non-beaded case. Here, multiplication by

\scriptstyle{W_{ss}(t)}$$\scriptstyle{s}$$\scriptstyle{s}

does not preserve the link relations, but the supplementary terms vanish thanks to the AS relation when applying \langle\,\cdot\,,\textrm{{exp}}_{\sqcup}(-\frac{1}{2}\raisebox{-18.20973pt}{ \leavevmode\hbox to28.87pt{\vbox to35.86pt{\pgfpicture\makeatletter\hbox{\hskip 2.6975pt\lower-7.11609pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{10.92093pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{10.92093pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{9.63092pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptstyle{W_{ss}(t)}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{11.38092pt}\pgfsys@lineto{0.0pt}{22.76186pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}{}\pgfsys@moveto{2.27614pt}{-2.27614pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.63551pt}{-3.78308pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptstyle{s}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}{{}}{}\pgfsys@moveto{2.27614pt}{23.90001pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.63551pt}{22.39307pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptstyle{s}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})\rangle.

The second point follows from Lemma 3.3. ∎

3.4 Categories of diagrams

For $\bar{\mathcal{A}}=\mathcal{A},\ \widetilde{\mathcal{A}}_{R},\textrm{ or }\widetilde{\mathcal{A}}^{\textrm{{w}}}_{R}$, define a category $\bar{\mathcal{A}}$ whose objects are associative words in the letters $(+,-)$ and whose set of morphisms are $\bar{\mathcal{A}}(v,u)=\oplus_{X}\bar{\mathcal{A}}(X)$, where $X$ runs over all compact oriented 1–manifolds with boundary identified with the set of letters of $u$ and $v$, with the following sign convention: for $u$, a $+$ when the orientation of $X$ goes towards the boundary point and a $-$ when it goes backward, and the converse for $v$. Composition is given by vertical juxtaposition, where the label of the created edges in the case of beaded or winding diagrams is defined with the same rule as in the definition of $\langle D,E\rangle$. The tensor product given by disjoint union defines a strict monoidal structure on $\bar{\mathcal{A}}$.

We finally define the target category of our extended Kricker invariant.

Notation*.*

Given a positive integer $g$ and a symbol $\natural$, set $\lfloor g\rceil^{\natural}=\{1^{\natural},\dots,g^{\natural}\}$. Set $\lfloor 0\rceil^{\natural}=\varnothing$.

Definition 3.7**.**

Fix non-negative integers $f$ and $g$. An $R$–beaded Jacobi diagram on $(\varnothing,\lfloor g\rceil^{+}\cup\lfloor f\rceil^{-})$ is top–substantial if it is $\lfloor g\rceil^{+}$–substantial.

Given two such diagrams $D$ and $E$, define their composition $D\circ E$ as the sum of all ways of gluing all $i^{+}$–labelled vertices of $D$ with all $i^{-}$–labelled vertices of $E$, fixing the orientations and labels of the created edges as in the definition of $\langle D,E\rangle_{S}$. We get a category ${}^{ts}\widetilde{\mathcal{A}}$ whose objects are non-negative integers, with set of morphisms from $g$ to $f$ the subspace of $\widetilde{\mathcal{A}}_{\mathbb{Q}(t)}(*_{\lfloor g\rceil^{+}\cup\lfloor f\rceil^{-}})$ generated by top-substantial diagrams. The identity of $g$ is \textrm{{exp}}_{\sqcup}(\sum_{i=1}^{g}\raisebox{-9.90276pt}{ \leavevmode\hbox to12.59pt{\vbox to29.37pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-6.0719pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{-2.73889pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptstyle{i^{-}}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{14.18016pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptstyle{i^{+}}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}) —the sum is trivial if $g=0$. The tensor product defined by disjoint union of diagrams provides ${}^{ts}\widetilde{\mathcal{A}}$ a strict monoidal structure.

4 Winding matrices

In this section, we define winding matrices associated with tangles with disks and bottom-top tangles with paths and interpret them as equivariant linking matrices in the case of bottom-top tangles. They will be useful in expressions of our invariant and splitting formulas.

4.1 First definition

We first define winding matrices in tangles with disks. Let $(\gamma,k)$ be a tangle with disks $d_{\ell}$. Write $\gamma$ as the disjoint union of components $\gamma_{i}$ for $i=1,\dots,n$. Fix a diagram of $(\gamma,k)$ and a base point $\star_{i}$ for each closed component $\gamma_{i}$ far from the crossings and the disks. Define the associated winding $w(\gamma_{i},\gamma_{j})\in\mathbb{Z}[t^{\pm 1}]$ of $\gamma_{i}$ and $\gamma_{j}$ in the following way. For a crossing $c$ between $\gamma_{i}$ and $\gamma_{j}$, denote $\varepsilon_{ij}(c)$ the algebraic intersection number of the union of the disks $d_{\ell}$ with the path that goes from $\star_{i}$, or the origin of $\gamma_{i}$, to $c$ along $\gamma_{i}$ and then from $c$ to $\star_{j}$, or the end-point of $\gamma_{j}$, along $\gamma_{j}$. If $i=j$, change component at the first occurence of $c$. Set

[TABLE]

where the sums are over all crossings between $\gamma_{i}$ and $\gamma_{j}$. Note that $w(\gamma_{j},\gamma_{i})(t)=w(\gamma_{i},\gamma_{j})(t^{-1})$. Now let $I$ and $J$ be two subsets of $\{1,\dots,n\}$ and denote by $\gamma_{I}$ and $\gamma_{J}$ the corresponding subtangles of $\gamma$. The winding matrix $W_{\gamma_{I}\gamma_{J}}$ associated with the fixed diagram and base points is the matrix whose coefficients are the windings $w(\gamma_{i},\gamma_{j})$ for $i\in I$ and $j\in J$ —denote it $W_{\gamma_{I}}$ when $I=J$. In this latter case, note that $W_{\gamma_{I}}$ is hermitian.

Lemma 4.1**.**

The winding matrix is invariant by isotopies which do not allow the base points to pass through the disks of the tangle. In particular, when $\gamma$ contains no closed components, it is an isotopy invariant.

Proof.

First note that the winding matrix is preserved when a crossing passes through a disk $d_{\ell}$. It is also preserved when a base point of a closed component passes through a crossing since the algebraic intersection number of this component with the union of the disks $d_{\ell}$ is trivial. Hence it only remains to check invariance with respect to framed Reidemeister moves performed far from the base points and the disks, which is direct. ∎

To completely understand the effect of an isotopy on the winding matrix $W_{\gamma_{I},\gamma_{J}}$, we shall describe its modification when a base point passes through a disk of the tangle. Fix a closed component $\gamma_{i}$. Fix a diagram of $(\gamma,k)$ with the base point of $\gamma_{i}$ located “just before” a disk $d_{\ell}$ of the tangle, as shown in the first part of Figure 20.

Consider another diagram of $(\gamma,k)$ which differs from the previous one only by the position of the base point $\star_{i}$, which is as shown on the second part of Figure 20. Let $\varepsilon=\pm 1$ give the sign of the intersection of $d_{\ell}$ and $\gamma_{i}$ which the base point passes through. It is easily seen that the winding matrix of the latter diagram is obtained from the winding matrix of the previous one by multiplication on the left by $T_{i}(t^{-\varepsilon})$ if $i\in I$ and on the right by $T_{i}(t^{\varepsilon})$ if $i\in J$, where $T_{i}(t)$ is the diagonal matrix whose diagonal coefficients are all 1 except a $t$ at the $i^{th}$ position.

We now define winding matrices for bottom-top tangles in $\mathbb{Q}$–cubes. Let $(B,K,\gamma)$ be a bottom-top tangle with paths in a $\mathbb{Q}$–cube. Let $(([-1,1]^{3},\Xi,\eta),L)$ be a surgery presentation of $(B,K,\gamma)$. Denote $(\eta,k)$ the associated tangle with disks. Fix a diagram of $([-1,1]^{3},\Xi,\eta)$ and a base point $\star_{i}$ for each component $L_{i}$ of $L$. Define the winding matrix of $(B,K,\gamma)$, with coefficients in $\mathbb{Q}(t)$, as:

[TABLE]

Note that $W_{L}(1)$ is the linking matrix of $L$, thus $W_{L}(1)\neq 0$ since $B$ is a homology cube. Hence $W_{L}$ is invertible over $\mathbb{Q}(t)$.

Lemma 4.2**.**

The winding matrix $W_{\gamma}$ is an isotopy invariant of $(B,K,\gamma)$.

Proof.

First, when the surgery presentation is fixed, the discussion of the previous subsection implies that the winding matrix does not depend on the choice of diagram and base points. Then independance with respect to Kirby moves is easily checked. We detail the less direct, which is the case when a surgery link component is added to a tangle component. Denote $\eta^{\prime}$ the tangle obtained from $\eta$ by adding the surgery component $L_{j}$ to $\eta_{i}$. We have:

[TABLE]

where $P$ is the $|L|\times|\eta|$ matrix whose only non-trivial term is a 1 at the $(j,i)$ position. ∎

4.2 Topological interpretation

Given a bottom-top tangle with paths in a $\mathbb{Q}$–cube, or a tangle with disks defined from a bottom-top tangle with paths in $[-1,1]^{3}$ with a surgery link, close the components of the tangle, say $\gamma$, by line segments on the top and bottom surfaces to get a link $\bar{\gamma}$. This provides well-defined linking numbers for the tangle components. If there is no path or disk, the winding matrix is the linking matrix. It is clear when working in $[-1,1]^{3}$. For bottom-top tangles in $\mathbb{Q}$–cubes, apply the following easy fact, which can be proved by adapting the proof of Proposition 4.5.

Fact 4.3*.*

Let $L$ be an oriented framed link in $[-1,1]^{3}$ whose linking matrix $\textrm{{Lk}}(L)$ is non-degenerate. Let $\xi$ and $\zeta$ be disjoint oriented knots in $[-1,1]^{3}\setminus L$ and denote $\xi^{\prime},\zeta^{\prime}$ the copies of $\xi,\zeta$ in the $\mathbb{Q}$–cube obtained from $[-1,1]^{3}$ by surgery on $L$. Then:

[TABLE]

More generally, when there are paths or disks, the winding matrix evaluated at $t=1$ is the linking matrix. We shall give a similar interpretation for the winding matrix at a generic $t$.

Let $(B,K,\gamma)$ be a bottom-top tangle in a $\mathbb{Q}$–cube. Let $E$ be the exterior of $K$ in $B$ and let $\tilde{E}$ be its covering associated with the kernel of the map $\pi_{1}(E)\to\mathbb{Z}=\langle t\rangle$ which sends the positive meridians of the components of $K$ to $t$. The automorphism group of the covering is isomorphic to $\mathbb{Z}$; let $\tau$ be the generator associated with the action of the positive meridians. Let $\zeta$ be a knot in $\tilde{E}$ such that there are a rational 2–chain $\Sigma$ in $\tilde{E}$ and $P\in\mathbb{Q}[t^{\pm 1}]$ which satisfy $\partial\Sigma=P(\tau)(\zeta)$. Let $\xi$ be another knot in $\tilde{E}$ such that the projections of $\zeta$ and $\xi$ in $E$ are disjoint. Define the equivariant linking number of $\zeta$ and $\xi$ as:

[TABLE]

The equivariant linking number is well-defined since $H_{2}(\tilde{E},\mathbb{Q})=0$ (see for instance [Mou14, Lemma 2.1]) and satisfies $\textrm{{lk}}_{e}(\tau(\zeta),\xi)=t\,\textrm{{lk}}_{e}(\zeta,\xi)$.

First consider a tangle with disks $(\gamma,k)$, with disks $d_{\ell}$ associated to the integer $k$, defined from a surgery presentation of a bottom-top tangle with paths in a $\mathbb{Q}$–cube, so that the closure $\bar{\gamma}_{i}$ of each component $\gamma_{i}$ is well-defined. Fix a diagram of $(\gamma,k)$ and base points $\star_{i}$ of its components. For an interval component, choose the base point to be its origin. Set $d=\cup_{\ell=1}^{k}d_{\ell}$. Let $E$ be the exterior of $\partial d$ in $[-1,1]^{3}$ and let $\tilde{E}$ be the infinite cyclic covering defined above. Let $\tilde{E}_{0}\subset\tilde{E}$ be a copy of the exterior of $d$ in $[-1,1]^{3}$. Define the lift $\tilde{\gamma_{i}}$ of $\bar{\gamma}_{i}$ in $\tilde{E}$ by lifting $\star_{i}$ in $\tilde{E}_{0}$. Given two subtangles $\gamma_{I}$ and $\gamma_{J}$ of $\gamma$, define the equivariant linking matrix $\textrm{{Lk}}_{e}(\tilde{\gamma}_{I},\tilde{\gamma}_{J})$ of their lifts with the equivariant linking numbers of the $\tilde{\gamma_{i}}$.

Lemma 4.4**.**

$W_{\gamma_{I}\gamma_{J}}=\textrm{{Lk}}_{e}(\tilde{\gamma}_{I},\tilde{\gamma}_{J})$**

Proof.

Set $\tilde{E}_{\ell}=\tau^{\ell}(\tilde{E}_{0})$, where $\tau$ is the generator of the automorphism group of the covering $\tilde{E}$ which corresponds to the action of the positive meridians of the $\partial d_{\ell}$. Fix $i\in I$ and $j\in J$. Since $\bar{\gamma}_{i}$ is null-homotopic in $[-1,1]^{3}\setminus\partial d$, it bounds a disk $D$ immersed in $[-1,1]^{3}\setminus\partial d$. Let $\tilde{D}$ be the lift of $D$ obtained by lifting $\star_{i}$ in $\tilde{E}_{0}$. Set $\tilde{D}_{\ell}=\tilde{D}\cap\tilde{E}_{\ell}$ and let $D_{\ell}$ be the image of $\tilde{D}_{\ell}$ in $E$. Set $c_{\ell}=\partial D_{\ell}$ and $\tilde{c}_{\ell}=\partial\tilde{D}_{\ell}$. Similarly, define the $c^{\prime}_{\ell}$ and $\tilde{c}^{\prime}_{\ell}$ from $\bar{\gamma}_{j}$. Assume the $c^{\prime}_{\ell}$ do not meet the $D_{\ell}$ along the disks of the tangle.

Thanks to Lemma 4.1, we have $w(\gamma_{i},\gamma_{j})=\sum_{\ell,\ell^{\prime}\in\mathbb{Z}}w(c_{\ell},c^{\prime}_{\ell^{\prime}})t^{\ell-\ell^{\prime}}$ for any choice of base points of the $c_{\ell}$ and $c^{\prime}_{\ell^{\prime}}$. Since these latter curves do not cross the disks of the tangle, we have $w(c_{\ell},c^{\prime}_{\ell^{\prime}})=\textrm{{lk}}(c_{\ell},c^{\prime}_{\ell^{\prime}})=\langle D_{\ell},c^{\prime}_{\ell^{\prime}}\rangle$, thus $w(\gamma_{i},\gamma_{j})=\sum_{\ell,\ell^{\prime}\in\mathbb{Z}}\langle D_{\ell},c^{\prime}_{\ell^{\prime}}\rangle t^{\ell-\ell^{\prime}}$. Lifting both $D_{\ell}$ and $c^{\prime}_{\ell^{\prime}}$ in $\tilde{E}_{\ell}$ does not change their algebraic intersection number, hence

[TABLE]

where the third equality holds since $\tau^{\ell^{\prime}}(\tilde{c}^{\prime}_{\ell-\ell^{\prime}})=\tau^{\ell^{\prime}}(\tilde{\gamma_{j}})\cap\tilde{E}_{\ell}$. ∎

Now consider a bottom-top tangle with paths in a $\mathbb{Q}$–cube $(B,K,\gamma)$. Since $\gamma$ is null-homotopic in $B\setminus K$, we have a well-defined equivariant linking matrix $\textrm{{Lk}}_{e}(\tilde{\gamma})$. Here, all components are intervals, so we have a canonical choice of base points.

Proposition 4.5**.**

Let $(B,K,\gamma)$ be a bottom-top tangle with paths in a $\mathbb{Q}$–cube. Then $W_{\gamma}=\textrm{{Lk}}_{e}(\tilde{\gamma})$.

Proof.

Let $(([-1,1]^{3},\Xi,\eta),L)$ be a surgery presentation of $(B,K,\gamma)$. Fix a diagram of $([-1,1]^{3},\Xi,\eta\cup L)$ and base points for the components of $L=\sqcup_{1\leq i\leq n}L_{i}$. Let $\tilde{E}$ be the infinite cyclic covering of the exterior of $\Xi$ in $[-1,1]^{3}$. Let $d$ be the disjoint union of disks in $[-1,1]^{3}$ bounded by $\hat{\Xi}$. Let $\tilde{L}=\sqcup_{1\leq i\leq n}\tilde{L}_{i}$, $\tilde{\gamma}$ and $\tilde{\eta}$ be the lifts of $L$, $\gamma$ and $\eta$ in $\tilde{E}$ with all base points in the same copy in $\tilde{E}$ of the exterior of $d$ in $[-1,1]^{3}$. We have to prove that:

[TABLE]

The infinite cyclic covering $\tilde{E}^{\prime}$ of the exterior of $K$ in $B$ is obtained from $\tilde{E}$ by surgery on $\cup_{\ell\in\mathbb{Z}}\tau^{\ell}(\tilde{L})$. Let $\tilde{\eta}_{x}$ and $\tilde{\eta}_{y}$ be components of $\tilde{\eta}$, and let $\tilde{\gamma}_{x}$ and $\tilde{\gamma}_{y}$ be the corresponding components of $\tilde{\gamma}$. For any knot $\lambda$ in $\tilde{E}$ or $\tilde{E}^{\prime}$, denote $m(\lambda)$ a positive meridian. For $1\leq i\leq n$, let $c_{i}$ be the parallel of $\tilde{L}_{i}$ which bounds a disk after surgery. In the group $H_{1}\left(\tilde{E}\setminus\cup_{\ell\in\mathbb{Z}}\left(\tau^{\ell}(\tilde{L})\cup\tau^{\ell}(\tilde{\eta}_{y})\right);\mathbb{Z}\right)$, we have

[TABLE]

and

[TABLE]

where multiplication by $t$ is given by the action of $\tau$ in homology. In $H_{1}(\tilde{E}^{\prime}\setminus\cup_{\ell\in\mathbb{Z}}\tau^{\ell}(\tilde{\gamma}_{y});\mathbb{Z})$, this gives $\tilde{\gamma}_{x}=\textrm{{lk}}_{e}(\tilde{\eta}_{x},\tilde{\eta}_{y})m(\tilde{\gamma}_{y})-\textrm{{Lk}}_{e}(\tilde{\eta}_{x},\tilde{L})\textrm{{Lk}}_{e}(\tilde{L})^{-1}\textrm{{Lk}}_{e}(\tilde{L},\tilde{\eta}_{y})m(\tilde{\gamma}_{y}).$ ∎

It is easily checked that the null LP–surgery defined in the introduction provides a move on the set of bottom-top tangles with paths in $\mathbb{Q}$–cubes, which corresponds to the move of null LP–surgery on Lagrangian cobordisms defined at the end of Subsection 2.1 via the map $D$ of Proposition 2.1. Moreover:

Corollary 4.6**.**

The winding matrix of a bottom-top tangle with paths in a $\mathbb{Q}$–cube is invariant under null LP–surgeries.

Proof.

Let $(B,K,\gamma)$ be a bottom-top tangle with paths in a $\mathbb{Q}$–cube associated with a Lagrangian cobordism with paths $(M,K)$. Let $\mathbf{C}$ be a null LP–surgery on $(M,K)$ made of a single $\mathbb{Q}$SK–pair. Let $(M^{\prime},K^{\prime})$ be the Lagrangian cobordism with paths obtained by surgery and let $(B^{\prime},K^{\prime},\gamma^{\prime})$ be the associated bottom-top tangle with paths in a $\mathbb{Q}$–cube. Let $\tilde{E}$ be the infinite cyclic covering of the exterior of $K$ in $B$. The nullity condition implies that the preimage of the $\mathbb{Q}$–handlebody $C$ is the disjoint union of $\mathbb{Q}$–handlebodies $C_{\ell}$ homeomorphic to $C$. The infinite cyclic covering $\tilde{E}^{\prime}$ of the exterior of $K^{\prime}$ in $B^{\prime}$ is obtained from $\tilde{E}$ by null LP–surgeries on all the $C_{\ell}$. This concludes since LP–surgeries preserve linking numbers (see for instance [Mou15, Lemma 2.1]). ∎

5 Construction of an LMO functor on $\widetilde{\mathcal{LC}ob}$

5.1 The functor $Z^{\bullet}:\widetilde{\mathcal{T}}_{q}\to\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}[t^{\pm 1}]}$

The definition of the functor $Z^{\bullet}:\widetilde{\mathcal{T}}_{q}\to\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}[t^{\pm 1}]}$ is based on the functor $Z:\mathcal{T}_{q}\to\mathcal{A}$ of [CHM08], which is a renormalization of the Le–Murakami functor [LM95, LM96]. We recall in Figure 21 the definition of $Z$ on the elementary $q$–tangles, where $\nu\in\mathcal{A}(\raisebox{-0.86108pt}{ \leavevmode\hbox to9.51pt{\vbox to9.51pt{\pgfpicture\makeatletter\hbox{\hskip 4.75253pt\lower-4.75253pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@curveto{4.55254pt}{2.51433pt}{2.51433pt}{4.55254pt}{0.0pt}{4.55254pt}\pgfsys@curveto{-2.51433pt}{4.55254pt}{-4.55254pt}{2.51433pt}{-4.55254pt}{0.0pt}\pgfsys@curveto{-4.55254pt}{-2.51433pt}{-2.51433pt}{-4.55254pt}{0.0pt}{-4.55254pt}\pgfsys@curveto{2.51433pt}{-4.55254pt}{4.55254pt}{-2.51433pt}{4.55254pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{4.55254pt}{0.85355pt}\pgfsys@lineto{4.55254pt}{0.67792pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{4.55254pt}{0.67792pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,)\cong\mathcal{A}(\,\raisebox{-2.15277pt}{ \leavevmode\hbox to0.4pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{10.92093pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{10.92093pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,)$ is the value of the Kontsevich integral on the zero framed unknot, $\Phi\in\mathcal{A}(\,\raisebox{-2.15277pt}{ \leavevmode\hbox to0.4pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.45999pt}\pgfsys@lineto{0.0pt}{11.38092pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{0.45999pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,\,\raisebox{-2.15277pt}{ \leavevmode\hbox to0.4pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.45999pt}\pgfsys@lineto{0.0pt}{11.38092pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{0.45999pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,\,\raisebox{-2.15277pt}{ \leavevmode\hbox to0.4pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.45999pt}\pgfsys@lineto{0.0pt}{11.38092pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{0.45999pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,)$ is a Drinfeld associator with rational coefficients and $\Delta^{+++}_{u_{1},u_{2},u_{3}}:\mathcal{A}(\,\raisebox{-2.15277pt}{ \leavevmode\hbox to0.4pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.45999pt}\pgfsys@lineto{0.0pt}{11.38092pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{0.45999pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,\,\raisebox{-2.15277pt}{ \leavevmode\hbox to0.4pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.45999pt}\pgfsys@lineto{0.0pt}{11.38092pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{0.45999pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,\,\raisebox{-2.15277pt}{ \leavevmode\hbox to0.4pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.45999pt}\pgfsys@lineto{0.0pt}{11.38092pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{0.45999pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,)\to\mathcal{A}(\,\raisebox{-2.15277pt}{ \leavevmode\hbox to0.4pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.45999pt}\pgfsys@lineto{0.0pt}{11.38092pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{0.45999pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,_{u_{1}u_{2}u_{3}})$ is obtained by applying $(|u_{i}|-1)$ times $\Delta$ on the $i$-th factor.

Let $(\gamma,k)$ be a $q$–tangle with disks. Assume $\gamma$ is transverse to $[-1,1]^{2}\times\{h_{i}(k)\}$ for all $i\in\{1,\dots,k\}$, and write $\gamma$ as a composition of $q$–tangles $\gamma_{i}$ by cutting along these levels, see Figure 22.

Write the bottom word of $\gamma_{i}$ as $w_{b}(\gamma_{i})=(v_{i})(w_{i})$, where $w_{i}$ corresponds to the part of the tangle which meets the disk $d_{i}$. Set:

[TABLE]

where $I_{v}$ is the identity on the word $v$ and $G_{v}$ is obtained from $I_{v}$ by adding a label $t$ (resp. $t^{-1}$) on skeleton components associated with a $-$ sign (resp. a $+$ sign), see Figure 23.

At the level of objects, $Z^{\bullet}$ forgets the parentheses. Invariance with respect to isotopy and to the cutting of $\gamma$ is due to invariance of the functor $Z$ and the following observation of Kricker [Kri00, Lemma 3.2.4].

Lemma 5.1**.**

For a winding Jacobi diagram $D\in\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}[t^{\pm 1}]}(w,v)$, we have $G_{v}\circ D=D\circ G_{w}$.

Proof.

Apply the relations Hol and Hol${}_{\textrm{w}}$ at all vertices of the diagram. ∎

Furthermore, $Z^{\bullet}$ is a clearly a functor and it preserves the tensor product on $\mathcal{T}_{q}\otimes\widetilde{\mathcal{T}}_{q}$ since $Z$ is tensor-preserving.

Lemma 5.2**.**

For any $q$–tangle with disks $(\gamma,k)$, $Z^{\bullet}(\gamma,k)$ is group-like.

Proof.

The fact that $Z(\gamma)$ is group-like for a $q$–tangle $\gamma$ follows from [LM97, Theorem 5.1]. This concludes since the $G_{v}$ are obviously group-like and the coproduct commutes with the composition. ∎

5.2 The functor $Z:\widetilde{\mathcal{T}}_{q}\mathcal{C}ub\to\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}(t)}$

The next step is to evaluate $Z^{\bullet}$ on the surgery presentation of a $q$–tangle with paths in a $\mathbb{Q}$–cube. Let $(B,K,\gamma)\in\widetilde{\mathcal{T}}_{q}\mathcal{C}ub(w,v)$. Let $(([-1,1]^{3},\Xi,\eta),L)$ be a surgery presentation of $(B,K,\gamma)$. The trivial link $\hat{\Xi}$ is the union of the boundaries of the disks $d_{i}=[0,1]\times[-1,1]\times\{h_{i}(k)\}$, where $k$ is the number of components of $\Xi$. Hence we have a $q$–tangle with disks $(\eta\cup L,k)$ and $Z^{\bullet}(\eta\cup L,k)\in\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}[t^{\pm 1}]}(\eta\cup L)$. Set:

[TABLE]

where the connected sum means that a copy of $\nu$ is summed to each component of $L$. Note that $Z^{\circ}((\Xi,\eta),L)$ is group-like since $Z^{\bullet}(\eta\cup L,k)$ and $\nu$ are group-like and $\chi_{\pi_{0}(L)}$ preserves the coproduct.

We want to apply formal Gaussian integration to $Z^{\circ}((\Xi,\eta),L)$. We work with a lift $\overline{Z^{\circ}((\Xi,\eta),L)}\in\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}[t^{\pm 1}]}(\eta,*_{\pi_{0}(L)})$. Fix a diagram of the $q$–tangle with disks $(\eta\cup L,k)$ transverse to the levels $\{h_{i}(k)\}$, and fix base points $\star_{i}$ on each component $L_{i}$ of $L$. Construct $\overline{Z^{\circ}((\Xi,\eta),L)}$ following the construction from the beginning of Section 5 for this diagram, with the skeleton components corresponding to the components of $L$ defined as intervals by cutting each component $L_{i}$ at the base point $\star_{i}$.

Lemma 5.3**.**

The lift $\overline{Z^{\circ}((\Xi,\eta),L)}$ is group-like, and we have:

[TABLE]

where $W_{L}$ is the winding matrix associated with our choice of diagram and base points and $H$ is $\pi_{0}(L)$–substantial.

Proof.

Check as in Lemma 5.2 that $\overline{Z^{\circ}((\Xi,\eta),L)}$ is group-like. We have to compute the part of $\overline{Z^{\circ}((\Xi,\eta),L)}$ made of $\pi_{0}(L)$–struts. We work with $\chi_{\pi_{0}(\eta)}^{-1}(\overline{Z^{\circ}((\Xi,\eta),L)})\in\widetilde{\mathcal{A}}_{\mathbb{Q}[t^{\pm 1}]}(*_{\pi_{0}(\eta)\cup\pi_{0}(L)})$, which is also group-like, in order to have a Hopf algebra structure on our diagram space. In particular, the group-like property implies that $\chi_{\pi_{0}(\eta)}^{-1}(\overline{Z^{\circ}((\Xi,\eta),L)})$ is the exponential of a series of connected diagrams. Since $\nu$ and the associator $\Phi$ have no terms with exactly two vertices, the only contributions to the $\pi_{0}(L)$–struts part come from the crossings between components of $L$. For $i\neq j$, the definition of $Z$ and the Hol${}_{\textrm{w}}$ relation show that the contribution of a crossing $c$ between $L_{i}$ and $L_{j}$ is \chi_{\pi_{0}(\eta)}^{-1}\raisebox{-3.44444pt}{{{\Huge(}}}\frac{1}{2}\textrm{sg}(c)\raisebox{-28.45274pt}{ \leavevmode\hbox to47.2pt{\vbox to42.4pt{\pgfpicture\makeatletter\hbox{\hskip 6.40398pt\lower-13.74614pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}} {}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{27.99277pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{27.99277pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} {}{{}}{}{{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.07097pt}{-8.31633pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptstyle{L_{i}}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{} {}{}{}{}{}{}{{}}\pgfsys@moveto{34.14365pt}{0.0pt}\pgfsys@lineto{34.14365pt}{27.99277pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{34.14365pt}{27.99277pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} {}{{}}{}{{}}{}\pgfsys@moveto{34.14365pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{30.82361pt}{-8.31633pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptstyle{L_{j}}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@lineto{16.61183pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.61183pt}{14.22638pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }{}\pgfsys@moveto{17.07182pt}{14.22638pt}\pgfsys@lineto{34.14365pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}{}\pgfsys@moveto{17.07182pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.15254pt}{17.75938pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptstyle{t^{\varepsilon_{ij}(c)}}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}). Hence the contribution of all crossings between $L_{i}$ and $L_{j}$ is

\scriptstyle{L_{i}}$$\scriptstyle{L_{j}}$$\scriptstyle{(W_{L})_{ij}}$$=$$\scriptstyle{L_{j}}$$\scriptstyle{L_{i}}$$\scriptstyle{(W_{L})_{ji}} . For $i=j$, the contribution of a self-crossing of $L_{i}$ is:

[TABLE]

Summed over all self-crossings of $L_{i}$, we get as strut part:

[TABLE]

Hence $\chi_{\pi_{0}(\eta)}^{-1}(\overline{Z^{\circ}((\Xi,\eta),L)})=\textrm{{exp}}_{\sqcup}\left(\frac{1}{2}W_{L}\right)\sqcup H^{\prime}$ where $H^{\prime}\in\widetilde{\mathcal{A}}_{\mathbb{Q}[t^{\pm 1}]}(*_{\pi_{0}(\eta)\cup\pi_{0}(L)})$ is $\pi_{0}(L)$–substantial. Set $H=\chi_{\pi_{0}(\eta)}(H^{\prime})$. ∎

The matrix $W_{L}(1)$ is the linking matrix of the link $L$, hence it is the presentation matrix of the first homology group of a $\mathbb{Q}$–cube. Thus $\det(W_{L}(1))\neq 0$ and $\overline{Z^{\circ}((\Xi,\eta),L)}$ is a non-degenerate Gaussian. Lemma 3.6 implies:

Lemma 5.4**.**

The formal Gaussian integral $\int_{\pi_{0}(L)}\overline{Z^{\circ}((\Xi,\eta),L)}$ does not depend on the lift $\overline{Z^{\circ}((\Xi,\eta),L)}\in\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}[t^{\pm 1}]}(\eta,*_{\pi_{0}(L)})$ of $Z^{\circ}((\Xi,\eta),L)\in\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}[t^{\pm 1}]}(\eta,\textrm{\scriptsize\raisebox{0.2ex}{\textcircled{\raisebox{-1.3ex}{\normalsize*}}}⃝}_{\pi_{0}(L)})$.

This allows to set:

[TABLE]

Proposition 5.5**.**

Let $(B,K,\gamma)$ be a $q$–tangle with paths in a $\mathbb{Q}$–cube. Fix a surgery presentation $(([-1,1]^{3},\Xi,\eta),L)$ of $(B,K,\gamma)$. Then:

[TABLE]

where U_{\pm}=Z^{\circ}((\varnothing,\varnothing),\raisebox{-2.84544pt}{ \leavevmode\hbox to27.39pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 11.58092pt\lower-5.89046pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{{}{}{}{}}} {{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {} {}{}{{}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{3.14279pt}{-2.54767pt}{5.69046pt}{-5.69046pt}{5.69046pt}\pgfsys@curveto{-8.83325pt}{5.69046pt}{-11.38092pt}{3.14279pt}{-11.38092pt}{0.0pt}\pgfsys@curveto{-11.38092pt}{-3.14279pt}{-8.83325pt}{-5.69046pt}{-5.69046pt}{-5.69046pt}\pgfsys@curveto{-2.54767pt}{-5.69046pt}{0.0pt}{-3.14279pt}{0.0pt}{-0.45999pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{-0.45999pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{-1.9639pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptstyle{\pm 1}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}), defines a functor $Z:\widetilde{\mathcal{T}}_{q}\mathcal{C}ub\to\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}(t)}$ which preserves the tensor product on $\mathcal{T}_{q}\mathcal{C}ub\otimes\widetilde{\mathcal{T}}_{q}\mathcal{C}ub$.

Proof.

We have to check that $Z(B,K,\gamma)$ does not depend on the surgery presentation. Independance with respect to the orientation of the components of $L$ follows from the argument of [BNGRT02b, Proposition 3.1]. The normalization term $U_{+}^{-\sigma_{+}(L)}\sqcup U_{-}^{-\sigma_{-}(L)}$ ensures independance with respect to the KI move as usual. Independance with respect to the KII move mainly follows from [GK04, Section 5.4]. More precisely, the argument of [GK04, Theorem 4] adapts [LMMO99, Proposition 1] to relate the values of $\overline{Z^{\circ}((\Xi,\eta),L)}$ for surgery links that differ from each other by a KII move. Then [GK04, Lemma 5.6] shows that this implies the invariance of the formal Gaussian integral. As noted in [BNGRT02b, Section 5.1], the argument remains valid when a surgery component is added to a tangle component since [GK04, Lemma 5.6] uses integration along the surgery component. ∎

Restricting the functor $Z:\widetilde{\mathcal{T}}_{q}\mathcal{C}ub\to\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}(t)}$ to $q$–tangles in $\mathbb{Q}$–cubes with no path, one recovers the functor $Z:\mathcal{T}_{q}\mathcal{C}ub\to\mathcal{A}$ of [CHM08, Definition 3.16]. When $\gamma$ is a bottom-top tangle and $K=\varnothing$, $\chi_{\pi_{0}(\gamma)}^{-1}(Z(B,\varnothing,\gamma))$ is group-like and $\chi_{\pi_{0}(\gamma)}^{-1}(Z(B,\varnothing,\gamma))=\textrm{{exp}}_{\sqcup}(\textrm{{Lk}}(\gamma))\sqcup H$ for some substantial and group-like $H$ [CHM08, Lemma 3.17]. We generalize this in the next result.

Lemma 5.6**.**

For any bottom-top $q$–tangle with paths $(B,K,\gamma)$ where $B$ is a $\mathbb{Q}$–cube, $\chi_{\pi_{0}(\gamma)}^{-1}(Z(B,K,\gamma))$ is group-like and:

[TABLE]

for some substantial and group-like $H$.

Proof.

The fact that $Z(B,K,\gamma)$ is group-like follows from the same property for $U_{+}$, $U_{-}$ and $\overline{Z^{\circ}((\Xi,\eta),L)}$, and Theorem 3.4. It implies that $\chi_{\pi_{0}(\gamma)}^{-1}(Z(B,K,\gamma))$ is group-like since $\chi_{\pi_{0}(\gamma)}^{\phantom{-1}}$ preserves the coproduct. The same computation as in the proof of Lemma 5.3 gives:

[TABLE]

where $H^{\prime}$ is substantial —note that $\chi_{\pi_{0}(\gamma)}$ and $\chi_{\pi_{0}(\eta)}$ are essentially the same before and after surgery on $L$. Integrate along $\pi_{0}(L)$:

[TABLE]

∎

5.3 The functor $\tilde{Z}:\widetilde{\mathcal{LC}ob}_{q}\to{}^{ts}\widetilde{\mathcal{A}}$

In this section, we define a functor on Lagrangian $q$–cobordisms with paths by applying the invariant $Z$ on bottom-top $q$–tangles with paths in $\mathbb{Q}$–cubes. The invariant $Z$ is functorial on $q$–tangles but not on bottom-top $q$–tangles, due to the different composition laws. To deal with this, we introduce some specific elements $\boldsymbol{\top}_{g}\in{}^{ts}\widetilde{\mathcal{A}}(*_{\lfloor g\rceil^{+}\cup\lfloor g\rceil^{-}})$ following [CHM08, Sec. 4]. Set:

[TABLE]

[TABLE]

[TABLE]

where the bottom-top tangle $T_{1}$ is drawn in Figure 9. As proven in [CHM08, Lemma 4.9]:

Lemma 5.7**.**

$\boldsymbol{\top}_{g}$* is a group-like element of $\widetilde{\mathcal{A}}_{\mathbb{Q}(t)}(*_{\lfloor g\rceil^{+}\cup\lfloor g\rceil^{-}})$ and $\boldsymbol{\top}_{g}=Id_{g}\sqcup H$ for some substantial and group-like $H$. In particular, $\boldsymbol{\top}_{g}$ is top-substantial and $\lfloor g\rceil^{-}$–substantial.*

Let $(M,K)$ be a Lagrangian $q$–cobordism with paths and denote $(B,K,\gamma)$ the associated bottom-top $q$–tangle with paths, of type $(g,f)$. We have $Z(B,K,\gamma)\in\widetilde{\mathcal{A}}^{\textrm{{w}}}_{\mathbb{Q}(t)}(\gamma)\cong\widetilde{\mathcal{A}}_{\mathbb{Q}(t)}(\gamma)$ and we consider $\chi^{-1}(Z(B,K,\gamma))\in\widetilde{\mathcal{A}}_{\mathbb{Q}(t)}(*_{\lfloor g\rceil^{+}\cup\lfloor f\rceil^{-}})$. It may not be top-substantial, but since $\boldsymbol{\top}_{g}$ is $\lfloor g\rceil^{-}$–substantial, we can set:

[TABLE]

At the level of objects, $\tilde{Z}$ sends a word on its number of letters. Direct adaptation of the proof of [CHM08, Lemma 4.10] implies that $\tilde{Z}$ preserves the composition and the next result follows, see [CHM08, Theorem 4.13].

Proposition 5.8**.**

$\tilde{Z}:\widetilde{\mathcal{LC}ob}_{q}\to{}^{ts}\widetilde{\mathcal{A}}$* is a functor which preserves the tensor product on $\mathcal{LC}ob_{q}\otimes\widetilde{\mathcal{LC}ob}_{q}$.*

Restricting the functor $\tilde{Z}:\widetilde{\mathcal{LC}ob}_{q}\to{}^{ts}\widetilde{\mathcal{A}}$ to Lagrangian $q$–cobordisms with no path, one recovers the functor $\tilde{Z}$ defined on $\mathcal{LC}ob_{q}$ in [CHM08, Theorem 4.13].

Lemmas 5.6 and 5.7 imply:

Lemma 5.9**.**

Let $(M,K)$ be a Lagrangian $q$–cobordism with paths and let $(B,K,\gamma)$ be the associated bottom-top $q$–tangle with paths. Then $\tilde{Z}(M,K)$ is group-like and $\tilde{Z}(M,K)=\textrm{{exp}}_{\sqcup}(W_{\gamma})\sqcup H$ for some substantial and group-like $H$.

5.4 Application to $\mathbb{Q}$SK–pairs

Let $(S,\kappa)$ be a $\mathbb{Q}$SK–pair. Let $M$ be the $\mathbb{Q}$–cube obtained from $S$ by removing the interior of a ball $B^{3}$ disjoint from $\kappa$. Isotoping $\kappa$ in $M$ and fixing a boundary parametrization $m$ of $M$, one can view $\kappa$ as the knot $\hat{K}$ associated with a Lagrangian cobordism with one path $(M,K)$. Since the top and bottom words are empty, we get a Lagrangian $q$–cobordism with one path.

Proposition 5.10**.**

Let $(S,\kappa)$ be a $\mathbb{Q}$SK–pair. Define as above an associated Lagrangian $q$–cobordism with one path $(M,K)$. Then $\tilde{Z}(S,\kappa)=\tilde{Z}(M,K)$ defines an invariant of $\mathbb{Q}$SK–pairs, which coincides with the Kricker invariant $Z^{\mathrm{rat}}$ for knots in $\mathbb{Z}$–spheres.

Proof.

When associating a cobordism with one path with a $\mathbb{Q}$SK–pair, we make a choice in the way we isotope the knot to the closure of a path. Once we work with a surgery presentation of our cobordism, this choice corresponds to the sweeping move represented in Figure 24.

But the right hand side diagram of this figure shows this move is trivial —as noted in [GK04, Lemma 3.26].

Coincidence with $Z^{\mathrm{rat}}$ is direct by construction. ∎

Remark*.*

The above proof does not work for a cobordism with more than one path, so we do not get an invariant of boundary links in $\mathbb{Q}$–spheres. One may obtain such an invariant by quotienting out the target space by suitable relations, see [GK04] for a construction of this kind.

Proposition 5.11**.**

Let $(S_{1},\kappa_{1})$ and $(S_{2},\kappa_{2})$ be $\mathbb{Q}$SK–pairs. The invariant $\tilde{Z}$ is given on their connected sum by:

[TABLE]

Proof.

As previously, associate Lagrangian $q$–cobordisms with one path $(M_{1},K_{1})$ and $(M_{2},K_{2})$ with $(S_{1},\kappa_{1})$ and $(S_{2},\kappa_{2})$ respectively. Construct a Lagrangian $q$–cobordism with one path $(M,K)$ associated with $(S,\kappa)=(S_{1},\kappa_{1})\sharp(S_{2},\kappa_{2})$ by stacking $(M_{1},K_{1})$ and $(M_{2},K_{2})$ together in the $y$ direction. Now $(M_{1},K_{1})$ and $(M_{2},K_{2})$ are obtained from the cube $[-1,1]^{3}$ with one disk by surgery on links $L_{1}$ and $L_{2}$ respectively. We obtain a surgery diagram for $(M,K)$ by drawing $L_{1}$ “in front” of $L_{2}$, or equivalently “around” $L_{2}$, see Figure 25. The result follows from this latter diagram since there is no crossing between $L_{1}$ and $L_{2}$.

∎

6 Splitting formulas

We first mention useful lemmas, namely [Mas15, Lemma 4.3] and [Mas15, Lemma 4.4]. Recall the tensor $\mu(\mathbf{C})$ was defined in the introduction.

Lemma 6.1** (Massuyeau).**

For a $\mathbb{Q}$–handlebody $C$ of genus $g$, there exists a boundary parametrization $c:\partial C_{0}^{g}\to C$ such that $(C,c)\in\mathcal{LC}ob(g,0)$.

Lemma 6.2** (Massuyeau).**

Let $\mathbf{C}=\left(\frac{C^{\prime}}{C}\right)$ be an LP–pair of genus $g$. Take boundary parametrizations $c:\partial C_{0}^{g}\to C$ and $c^{\prime}:\partial C_{0}^{g}\to C^{\prime}$ compatible with the fixed identification $\partial C\cong\partial C^{\prime}$ such that $(C,c)\in\mathcal{LC}ob(g,0)$ and $(C^{\prime},c^{\prime})\in\mathcal{LC}ob(g,0)$. Then:

[TABLE]

where $\tilde{Z}_{1}$ is the i–degree 1 part of $\tilde{Z}$ and $\mu(\mathbf{C})$ is considered as an element of $\widetilde{\mathcal{A}}_{\mathbb{Q}(t)}(*_{\lfloor g\rceil^{+}})$ via the inclusion $\Lambda^{3}H_{1}(\mathcal{C};\mathbb{Q})\hookrightarrow\widetilde{\mathcal{A}}_{\mathbb{Q}(t)}(*_{\lfloor g\rceil^{+}})$ defined by:

[TABLE]

Let $(M,K)\in\widetilde{\mathcal{LC}ob}_{q}(w,v)$. Let $\mathbf{C}=(\mathbf{C}_{1},\dots,\mathbf{C}_{n})$ be a null LP–surgery on $(M,K)$. Let $e_{i}$ be the genus of $C_{i}$. For $1\leq i\leq n$, take boundary parametrizations $c_{i}:\partial C_{0}^{e_{i}}\to C_{i}$ and $c_{i}^{\prime}:\partial C_{0}^{e_{i}}\to C_{i}^{\prime}$ compatible with the fixed identification $\partial C_{i}\cong\partial C_{i}^{\prime}$ such that $(C_{i},c_{i})\in\mathcal{LC}ob(e_{i},0)$ and $(C_{i}^{\prime},c_{i}^{\prime})\in\mathcal{LC}ob(e_{i},0)$. Set $e=\sum_{i=1}^{n}e_{i}$. Take a collar neighborhood $m_{-}(F_{f})\times[-1,\varepsilon-1]$ of the bottom surface $m_{-}(F_{f})$. Take pairwise disjoint solid tubes $T_{i}$, $i=1,\dots,n$, such that $T_{i}$ connects $(c_{i})_{-}(F_{0})$ to a disk in $m_{-}(F_{f})\times\{\varepsilon-1\}$ in the complement of the $C_{j}$’s, the collar neighborhood and $K$. This provides a decomposition of the cobordism $(M,K)$ as:

[TABLE]

where $f$ is the number of letters of $v$ (see Figure 26). It is proved in [Mas15, Section 4.4] that $N$ is a Lagrangian cobordism. The nullity condition on the surgery ensures that $\hat{J}$ is a boundary link. Thus $(N,J)$ is a Lagrangian cobordism with paths.

With the surgery $\mathbf{C}$ is associated the tensor $\mu(\mathbf{C})\in\mathcal{A}_{\mathbb{Q}}(H_{1}(\mathcal{C};\mathbb{Q}))$. Let $W$ be a square matrix of size $e$ with coefficients in $\mathbb{Q}(t)$. Interpret $W$ as a hermitian form on $H_{1}(\mathcal{C};\mathbb{Q})$ written in the basis $(([(c_{i})_{+}(\beta_{j})])_{1\leq j\leq e_{i}})_{1\leq i\leq n}$. Given an $H_{1}(\mathcal{C};\mathbb{Q})$–colored Jacobi diagram, one can glue some legs of the diagram with $W$, see Figure 3. Changing the labels of the univalent vertices via the bijection

[TABLE]

onto $\{1,\dots,e\}$, this provides a diagram in $\widetilde{\mathcal{A}}_{\mathbb{Q}(t)}(*_{\lfloor e\rceil^{+}})$.

The following result is a direct adaptation of [Mas15, Section 4.4], with the winding matrices playing the role of the linking matrices.

Proposition 6.3**.**

Let $(M,K)$ be a Lagrangian $q$–cobordism with paths and let $(B,K,\gamma)$ be the associated bottom-top $q$–tangle with paths. Let $\mathbf{C}=(\mathbf{C}_{1},\dots,\mathbf{C}_{n})$ be a null LP–surgery on $(M,K)$. Define as above a decomposition of the cobordism $(M,K)$. Choose top and bottom words for $(N,J)$ and the $(C_{i},\varnothing)$ in order to get a decomposition of the Lagrangian $q$–cobordism $(M,K)$ as $(M,K)=((C_{1},\varnothing)\otimes\dots\otimes(C_{n},\varnothing)\otimes Id_{v})\circ(N,J)$. Let $(D,J,\varsigma)$ be the bottom-top $q$–tangle with paths associated with $(N,J)$. Let $\varsigma^{\mathbf{c}}$ be the subtangle of $\varsigma^{-}$ corresponding to the $C_{i}$’s. Let $e$ be the number of components of $\varsigma^{\mathbf{c}}$. Let $\tilde{\rho}_{\mathbf{c}}:\widetilde{\mathcal{A}}_{\mathbb{Q}(t)}(*_{\lfloor e\rceil^{+}})\to\widetilde{\mathcal{A}}_{\mathbb{Q}(t)}(*_{\lfloor g\rceil^{+}\cup\lfloor f\rceil^{-}})$ be the linear form which changes the labels of the univalent vertices as follows:

[TABLE]

Then:

[TABLE]

where $\mathbf{C}_{I}=((\mathbf{C}_{i})_{i\in I})$ and $\equiv_{n}$ means “equal up to i–degree at least $n+1$ terms”.

Note that $W_{\gamma}=W_{\varsigma\setminus\varsigma^{\mathbf{c}}}$ by Proposition 4.6.

For a cobordism with one path, the next result gives a more intrinsic version of these formulas, which does not refer to a decomposition of the cobordism. A similar result is given by in [Mas15, Lemma 4.1] for a cobordism with no path.

Given a null LP–surgery $\mathbf{C}=(\mathbf{C}_{1},\dots,\mathbf{C}_{n})$ on a Lagrangian cobordism with paths $(M,K)$, define a hermitian form $\ell_{(M,K)}(\mathbf{C}):H_{1}(\mathcal{C};\mathbb{Q})\times H_{1}(\mathcal{C};\mathbb{Q})\to\mathbb{Q}(t)$ in the same way as $\ell_{(S,\kappa)}(\mathbf{C})$ was defined in the introduction. Also define a map $\rho_{\mathbf{c}}:\mathcal{A}_{\mathbb{Q}}(H_{1}(\mathcal{C};\mathbb{Q}))\to\widetilde{\mathcal{A}}_{\mathbb{Q}(t)}(*_{\lfloor g\rceil^{+}\cup\lfloor f\rceil^{-}})$ which changes the labels of the univalent vertices by first sending them in $H_{1}(M;\mathbb{Q})$ via $H_{1}(\mathcal{C};\mathbb{Q})\cong\oplus_{i=1}^{n}H_{1}(C_{i};\mathbb{Q})\to H_{1}(M;\mathbb{Q})$, and then writing them in terms of the $[m_{+}(\beta_{i})]$ and $[m_{-}(\alpha_{i})]$.

A direct adaptation of (the end of) [Mas15, Section 4.4] gives:

Proposition 6.4**.**

Let $(M,K)\in\widetilde{\mathcal{LC}ob}_{q}(w,v)$ be a Lagrangian $q$–cobordism with one path. Let $\mathbf{C}=(\mathbf{C}_{1},\dots,\mathbf{C}_{n})$ be a null LP–surgery on $(M,K)$. Let $(B,K,\gamma)$ be the bottom-top tangle with one path associated with $(M,K)$. Then:

[TABLE]

Proof of Theorem 1.1.

Use Propositions 5.10 and 6.4. The strut part disappears since we deal with a cobordism from $F_{0}$ to $F_{0}$. The map $\rho_{\mathbf{c}}$ kills all terms with at least one univalent vertex since a Lagrangian cobordism from $F_{0}$ to $F_{0}$ has trivial first homology group over $\mathbb{Q}$. ∎