On the functoriality of sl(2) tangle homology

Anna Beliakova; Matthew Hogancamp; Krzysztof Karol Putyra; Stephan; Martin Wehrli

arXiv:1903.12194·math.AT·June 14, 2023

On the functoriality of sl(2) tangle homology

Anna Beliakova, Matthew Hogancamp, Krzysztof Karol Putyra, Stephan, Martin Wehrli

PDF

Open Access

TL;DR

This paper establishes a functorial framework for sl(2) tangle homology by constructing explicit equivalences between web categories and cobordism categories, enabling strictly functorial link invariants.

Contribution

It introduces web versions of arc algebras and their covers, providing a functorial approach to link homology theories factoring through the Bar-Natan category.

Findings

01

Constructed an explicit equivalence between web and cobordism categories.

02

Defined web versions of arc algebras with quasi-hereditary covers.

03

Achieved a strictly functorial version of annular link homology.

Abstract

We construct an explicit equivalence between the (bi)category of gl(2) webs and foams and the Bar-Natan (bi)category of Temperley-Lieb diagrams and cobordisms. With this equivalence we can fix functoriality of every link homology theory that factors through the Bar-Natan category. To achieve this, we define web versions of arc algebras and their quasi-hereditary covers, which provide strictly functorial tangle homologies. Furthermore, we construct explicit isomorphisms between these algebras and the original ones based on Temperley-Lieb cup diagrams. The immediate application is a strictly functorial version of the Beliakova-Putyra-Wehrli quantization of the annular link homology.

Tables1

Table 1. Table 1: A basis for foams bounded by a circle, represented as planar diagrams and as foams.

Web	Completion	Basis
		$⟷ 0$ ,
		$⟷ 0$
		$⟷ = 0$ ,
		$⟷ = - 0$

Equations176

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H^{n} := a, b ⨁ Hom_{BN} (a, b),

H^{n} := a, b ⨁ Hom_{BN} (a, b),

W^{B} := a, b \in B ⨁ Hom_{Foam (Σ)} (a, b),

W^{B} := a, b \in B ⨁ Hom_{Foam (Σ)} (a, b),

h : K_{0} (A^{B, λ}) \otimes_{Z [q^{\pm 1}]} k \to qHH_{0} (A^{B, λ})

h : K_{0} (A^{B, λ}) \otimes_{Z [q^{\pm 1}]} k \to qHH_{0} (A^{B, λ})

Kh_{A_{q}} (T) := λ ⨁ qHH_{∙} (A^{B, λ}, C_{A} (T; λ))

Kh_{A_{q}} (T) := λ ⨁ qHH_{∙} (A^{B, λ}, C_{A} (T; λ))

= q + q^{- 1}

= q + q^{- 1}

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(ω, ω^{'}) := (q + q^{- 1})^{ℓ (- ω \cup ω^{'})},

(ω, ω^{'}) := (q + q^{- 1})^{ℓ (- ω \cup ω^{'})},

=

=

=

=

de g (S) := - χ (S_{b}) + 2 dots (S) .

de g (S) := - χ (S_{b}) + 2 dots (S) .

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de g (S) := (b - a) - χ (S_{b}) + 2 dots (S) + \frac{# Σ _{b}}{2},

de g (S) := (b - a) - χ (S_{b}) + 2 dots (S) + \frac{# Σ _{b}}{2},

\mathrm{Hom}_{{\textbf{Foam}}(\Sigma)}(\omega,\omega^{\prime})\cong\mathit{Foam}(-\omega\cup\omega^{\prime})\big{\{}\tfrac{\#\Sigma_{b}}{2}\big{\}}

\mathrm{Hom}_{{\textbf{Foam}}(\Sigma)}(\omega,\omega^{\prime})\cong\mathit{Foam}(-\omega\cup\omega^{\prime})\big{\{}\tfrac{\#\Sigma_{b}}{2}\big{\}}

Hom_{Foam (Σ)} (ω, ω^{'}) ∋ S ⟼ S^{!} \in Hom_{Foam (Σ)} (ω^{'}, ω)

Hom_{Foam (Σ)} (ω, ω^{'}) ∋ S ⟼ S^{!} \in Hom_{Foam (Σ)} (ω^{'}, ω)

A_{b} := k [X] / (X^{2} - h X - t) and A_{r} := k

A_{b} := k [X] / (X^{2} - h X - t) and A_{r} := k

Δ_{b} (1)

Δ_{b} (1)

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TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Geometric and Algebraic Topology

Full text

On the functoriality of $\mathfrak{sl}_{2}$ tangle homology

Anna Beliakova

Universität Zürich

Zürich, Switzerland

Matthew Hogancamp

University of Southern California

Los Angeles, CA

Krzysztof K. Putyra

Universität Zürich

Zürich, Switzerland

Stephan M. Wehrli

Syracuse University

Syracuse, NY

Abstract

We construct an explicit equivalence between the (bi)category of $\mathfrak{gl}_{2}$ webs and foams and the Bar-Natan (bi)category of Temperley–Lieb diagrams and cobordisms. With this equivalence we can fix functoriality of every link homology theory that factors through the Bar-Natan category. To achieve this, we define web versions of arc algebras and their quasi-hereditary covers, which provide strictly functorial tangle homologies. Furthermore, we construct explicit isomorphisms between these algebras and the original ones based on Temperley–Lieb cup diagrams. The immediate application is a strictly functorial version of the Beliakova–Putyra–Wehrli quantization of the annular link homology.

1 Introduction
1.1 The equivalence of foams and Bar-Natan cobordisms
1.2 Functorial tangle homology
1.3 Functoriality of quantized annular Khovanov homology
1.4 Further generalizations
1.5 Organization of the paper
1.6 Conventions and notation
1.7 Acknowledgments
2 Main players
2.1 Webs
2.2 Foams
2.3 Decategorification
2.4 Higher structures
2.5 Blanchet evaluation formula
3 Shadings and a basis of foams
3.1 Shadings and trivalent manifolds
3.2 Bicolored isotopies
3.3 Cup foams
4 Equivalences of foam and cobordism categories
4.1 Embedding cobordisms into foams
4.2 A coherent way to forget red facets
4.3 An equivalence of bicategories
4.4 Comparison of Khovanov brackets
5 A diagrammatic TQFT on ${\textbf{Foam}}(\emptyset)$ .
5.1 A planar representation of cup foams
5.2 Action of foams
6 The Blanchet–Khovanov invariant of tangles with balanced boundaries
6.1 A linear basis of webs
6.2 Blanchet–Khovanov algebras
6.3 Blanchet–Khovanov bimodules
6.4 A functorial homology for tangles with balanced boundaries
7 Subquotient algebras and an invariant for all tangles
7.1 Balancing
7.2 Webs and foams with platforms

1 Introduction

In 1999 Khovanov [KhHom] defined for any link in the 3-sphere a chain complex, whose homotopy type—hence, homology—is a link invariant and whose Euler characteristic is the Jones polynomial. It was later extended to tangles between even collections of points [KhArcAlgebras] and then to all tangles [ChenKhov, BrunStroII]. The main advantage of the Khovanov homology with respect to the Jones polynomial is that link cobordisms induce chain maps between Khovanov’s complexes [KhFunct, Jacobsson, DrorCob]. Even though the original construction is not strictly functorial—the sign of the chain map associated with a link cobordism depends on the decomposition of the cobordism into elementary pieces [Jacobsson]—it was used by Rasmussen to provide a lower bound for the slice genus of a knot and a combinatorial proof of the Milnor conjecture [SliceGenusBound].

In the last 15 years there were many attempts to fix the functoriality of Khovanov homology. In [ClarkMorrisonWalker, CaprauFoams, Vogel] this was done by modifying the Bar-Natan category [DrorCob] and enlarging the ground ring. In 2014 Blanchet [Blanchet] proposed a more elegant solution, which does not change the ring of scalars, but replaces circles and surfaces in the Bar-Natan category with webs and foams: certain planar trivalent graphs and singular cobordisms between them respectively. This construction, commonly referred to as $\mathfrak{gl}_{2}$ homology, has been widely accepted as the most natural way to fix functoriality of Khovanov homology. A priori potentially different, $\mathfrak{gl}_{2}$ homology coincides with Khovanov homology in case of links [Blanchet], but the case of tangles has been analyzed only partially by Ehrig, Stroppel and Tubbenhauer in [WebAlgebras].

The Hochschild homology of the Chen–Khovanov invariant of an $(n,n)$ -tangle $T$ has been identified in [QntHom] with the annular Khovanov homology of the annular closure $\widehat{T}$ of the tangle. In the same paper the annular invariant has been quantized by deforming the Hochschild homology. Our original goal was to make this quantized annular homology functorial, in order to construct its colored version following [KhColored] and [CooperKrushkal]. These quantized colored homologies are treated in the follow up paper [QntColored, ViennaTalk], where we also show that both complexes coincide when the deformation parameter is generic. In order to obtain a strictly functorial quantized annular homology, we wanted first to understand the Ehrig–Stroppel–Tubbenhauer isomorphism between Khovanov’s arc algebras and their web algebras, and then reconstruct the Chen–Khovanov functor in the framework of webs and foams. However, after a chain of simplifications of their arguments, especially replacing the foam basis used in [WebAlgebras] with another one, more natural from the topological perspective, we understood the real reason why all the isomorphisms popped out: foams and cobordisms constitute equivalent bicategories. By using a particularly nice basis of foams, we construct such a equivalence explicitly and use it to obtain a web versions of the TQFT functors from [KhArcAlgebras, ChenKhov, BrunStroII].

In the following sections we discuss the above in more details.

1.1 The equivalence of foams and Bar-Natan cobordisms

In order to compute Khovanov homology of a link $L$ , one first picks its diagram $D$ and constructs the cube of resolutions of $D$ : a commutative diagram in the shape of the $c$ -dimension cube, where $c$ counts crossings in $D$ , with vertices decorated by Kauffman resolutions of $D$ and edges by saddle cobordisms between them [KhHom]. Applying a 2-dimensional TQFT to this cube, changing signs of some maps, decorating edges, and collapsing the cube along diagonals results in an actual chain complex, which—depending on the choice of the TQFT functor—computes the Khovanov homology of $L$ or its deformation.

It was observed by Bar-Natan that most of the construction can be performed formally before applying a TQFT functor to get an invariant of a tangle $T$ in the form of a formal complex $\llbracket T\rrbracket$ called the Khovanov bracket of $T$ [DrorCob]. This complex is constructed in the Bar-Natan bicategory BN, the locally additive graded bicategory with objects collections of points on a line, 1-morphisms generated by flat tangles, and 2-morphisms generated by surfaces with dots modulo the following local relations:

•

sphere evaluations:

[TABLE]

•

neck cutting relation:

[TABLE]

•

dot reduction:

[TABLE]

Here $h$ and $t$ are fixed elements of the ring of scalars $\Bbbk$ . When $h=0$ , then the neck cutting relation evaluates a handle attached to a plane as a dot scaled by 2. Because of that it is common to think of a dot as ,,half” of a handle, even when 2 is not an invertible scalar. However, this interpretation is not correct if $h\neq 0$ , in particular in the universal case $\Bbbk=\mathbb{Z}[h,t]$ .

The formal bracket is projectively functorial [DrorCob]. Indeed, there is a way to associate a formal chain map with each Reidemeister move as well as any cobordism with a unique critical point. One constructs a formal chain map for any tangle cobordism by decomposing the cobordism into a sequence of the above elementary pieces and composing the associated maps; choosing a different decomposition may at most change the global sign of the map.

In Blanchet’s construction [Blanchet] the role of flat tangles is played by webs, trivalent graphs with each edge colored blue or red,111 When compared to [Blanchet], blue edges are those with label 1 and red edges are those with label 2.

and dotted surfaces are replaced with foams, which are singular cobordisms with each facet also colored blue or red. They constitute a bicategory Foam, where certain local relations between foams, including (1.1)–(1.3), are imposed (see Definition 2.6). Following [DrorCob] we can construct a formal complex $\llbracket T\rrbracket_{\mathrm{F}}$ in Foam, which we refer to as the Blanchet–Khovanov bracket.

The collection of blue edges of a web $\omega$ is a flat tangle $\omega_{b}$ , which we call the underlying tangle of $\omega$ . Likewise, there is an underlying surface $S_{b}$ associated with any foam $S$ . It is tempting to consider a 2-functor ${\textbf{Foam}}\to{\textbf{BN}}$ that forgets red edges in webs and red facets in foams. However, this operation is not compatible with relations between foams, and it is not clear at first how to solve this problem. For instance, it was observed in [KhViaHowe] that if such a functor exists, then it cannot be identity on all foams with no red facets.

We resolved the above problem by taking into account the orientation of blue edges and facets. Shortly speaking, we fix an orientation for each flat tangle and surface in a canonical way, reinterpreting them as webs and foams respectively (recall that tangles and surfaces from BN, though orientable, come with no particular orientation). This results in a 2-functor, which however does not reach every object of Foam. In order to fix this we replace BN with the product ${\textbf{wBN}}:={\textbf{BN}}\times\mathbb{Z}$ , where $\mathbb{Z}$ is seen as a discrete bicategory. We use the extra integer to determine how many red points, edges, or facets has to be added to the right of the oriented blue points, tangle, or surface respectively.222 Compare this with the relation between the weight lattices of $\mathfrak{sl}_{2}$ and $\mathfrak{gl}_{2}$ —the latter is isomorphic to the product of the former with $\mathbb{Z}$ .

This way we end up with a 2-functor $\mathcal{E}\colon{\textbf{wBN}}\to{\textbf{Foam}}$ , such that every object of Foam is equivalent to one from the image of $\mathcal{E}$ .

{PreTheoremA}

The 2-functor $\mathcal{E}\colon{\textbf{wBN}}\to{\textbf{Foam}}$ is an equivalence of bicategories.

From the point of view of representation theory, $\mathcal{E}$ and its inverse can be understood as the categorification of the induction–restriction pair between representations of $\mathfrak{sl}_{2}$ and $\mathfrak{gl}_{2}$ .

There is also a local version of Theorem A. Having fixed a collection $\Sigma$ of oriented blue and red points on $\partial\mathbb{D}^{2}$ , write ${\textbf{Foam}}(\Sigma)$ for the category of webs in $\mathbb{D}^{2}$ bounded by $\Sigma$ and foams in $\mathbb{D}^{2}\times[0,1]$ between such webs. Likewise we consider the category ${\textbf{BN}}(\Sigma_{b})$ of flat tangles bounded by $\Sigma_{b}$ and dotted surfaces between them, where $\Sigma_{b}$ is the collection of blue points from $\Sigma$ . We construct a functor $\mathcal{E}_{\Sigma}\colon{\textbf{BN}}(\Sigma_{b})\to{\textbf{Foam}}(\Sigma)$ in Section 4.1 by extending coherently all flat tangles to webs bounded by $\Sigma$ and surfaces to foams.

{PreTheoremB}

The functor $\mathcal{E}_{\Sigma}\colon{\textbf{BN}}(\Sigma_{b})\to{\textbf{Foam}}(\Sigma)$ is an equivalence of categories.

We construct the functor $\mathcal{E}_{\Sigma}$ explicitly as well as its inverse $\mathcal{E}_{\Sigma}^{\vee}$ . The latter not only forgets red facets of foams, but also scales them by a sign when necessary; we provide an explicit way to compute these signs in terms of the Blanchet evaluation of foams. When combined with a homological argument presented in [OddKh, ChCob], Theorem B implies that for every tangle $T$ the image of the Khovanov bracket $\llbracket T\rrbracket$ under $\mathcal{E}_{\Sigma}$ is isomorphic to the Blanchet–Khovanov bracket $\llbracket T\rrbracket_{\mathrm{F}}$ . Hence, any TQFT functor on ${\textbf{BN}}(\Sigma_{b})$ that leads to an invariant tangle or link homology can be precomposed with $\mathcal{E}_{\Sigma}^{\vee}$ to obtain a functor on ${\textbf{Foam}}(\Sigma)$ that computes the same homology groups, but which is strictly functorial with respect to tangle cobordisms.

1.1.1 Main tools: shadings and bicolored isotopies

The key step in the proofs of Theorems A and B is to understand how foams with the same underlying surface are related. We achieve this by constructing foams from shadings. A shading is a union of two possibly intersecting surfaces: a non-oriented blue and an oriented red one, that are in general position in $\mathbb{R}^{3}$ , together with a checkerboard black and white coloring of the connected components of their complement, called regions. Forgetting those red facets of a shading, the orientations of which disagree with the one induced from the white regions, results in a foam, and all foams can be constructed this way. The same applies to webs.

A particularly nice feature of representing foams by shadings is the flexibility of this construction, which we call the bicolored isotopy argument: deforming any of the two surfaces by an isotopy results in a foam that differs from the original one only up to a sign or replacing some dots with their duals (see Proposition 2.9 in Section 2.2 for a precise statement). This has a number of important consequences:

•

closed foams can be evaluated (Theorem 2.13) using the bicolored isotopy argument by moving the blue and red facets away from each other,

•

more generally, foams with the same boundary and underlying surfaces coincide up to a sign and types of dots (Proposition 2.9),

•

a foam, the underlying surface of which is a product $\omega\times[0,1]$ , is invertible.

We then use the above to construct a basis of the space of foams bounded by a closed web $\omega$ . It is given in terms of shadings of a plane that extends $\omega$ , the blue loops of which may carry dots. The foam associating with such a picture $\omega^{+}$ is given by attaching blue and red cups to the loops of $\omega^{+}$ —red cups above all blue ones—and placing a dot at the minimum of every blue cup attached to a loop that is marked by a dot. This leads to an explicit description of the tautological TQFT functor on ${\textbf{Foam}}(\emptyset)$ that associates the space $\mathrm{Hom}_{{\textbf{Foam}}(\emptyset)}(\emptyset,\omega)$ with a closed web $\omega$ , presented in Section 5. When compared with [WebAlgebras], our basis is not only easier to visualize, but also the formula for the action of foams involves less signs.

1.2 Functorial tangle homology

Khovanov extended his construction first to tangles with an even number of boundary points at each side [KhArcAlgebras]. For this he constructed a 2-functor $\mathcal{F}_{\!\textit{Kh}}^{\circ}\colon{\textbf{BN}}^{\circ}\to{\textbf{Bimod}}$ , where ${\textbf{BN}}^{\circ}$ is the subbicategory of BN with only even collections of points as objects. The 2-functor $\mathcal{F}_{\!\textit{Kh}}^{\circ}$ associates with a collection of $2n$ points the arc algebra

[TABLE]

where $a$ and $b$ run through the set of Temperley–Lieb cup diagrams in $\mathbb{R}\times(-\infty,0]$ with $2n$ boundary points at the top boundary line.333 This presentation of $H^{n}$ comes from [StableKh].

This algebra $H^{n}$ is known to categorify the invariant subspace $\mathrm{Inv}(V^{\otimes n})$ of $V^{\otimes n}$ , where $V$ is the fundamental representation of $\mathcal{U}_{q}(\mathfrak{sl}_{2})$ . Cup diagrams parametrize indecomposable projective $H^{n}$ -modules, which in turn correspond to elements of the canonical basis of $V^{\otimes n}$ . Let $\mathit{CKh}(T)$ be the chain complex associated with an $(2n,2n^{\prime})$ -tangle $T$ , i.e. the result of applying $\mathcal{F}_{\!\textit{Kh}}^{\circ}$ to $\llbracket T\rrbracket$ . The functors $\mathit{CKh}(T)\otimes(-)$ lift the action of tangles on $\mathrm{Inv}(V^{\otimes n})$ to the derived categories of the arc algebras [KhArcAlgebras].

In order to categorify the whole tensor power $V^{\otimes n}$ , Chen and Khovanov considered a family of algebras $A^{k,n-k}\!$ , where $0\leq k\leq n$ , each constructed as a subquotient of $H^{n}$ . These algebras were discovered independently by Stroppel [ParabolicO], who proved with Brundan than they are quasi-hereditary covers of arc algebras and Koszul [BrunStroI, BrunStroII]. Furthermore, projective modules over $A^{k,n-k}$ categorify the weight space $V^{\otimes n}(\lambda)$ with $\lambda=n-2k$ [ChenKhov, BrunStroI]. As in the case of arc algebras, there is a family of 2-functors $\mathcal{F}_{\!\textit{Kh}}^{\lambda}\colon{\textbf{BN}}\to{\textbf{Bimod}}$ , such that $\mathcal{F}_{\!\textit{Kh}}^{\lambda}$ assigns to a collection of $n$ points the algebra $A^{k,n-k}$ with $\lambda=n-2k$ [ChenKhov, BrunStroII]. Write $\mathit{CKh}(T;\lambda)$ for the result of applying $\mathcal{F}_{\!\textit{Kh}}^{\lambda}$ to $\llbracket T\rrbracket$ . Then the functor $\mathit{CKh}(T;\lambda)\otimes(-)$ lifts the action of $T$ on the weight space $V^{\otimes n}(\lambda)$ .

Using Theorem A we can contruct a strictly functorial version of both Khovanov and Chen–Khovanov homologies by precomposing $\mathcal{F}_{\!\textit{Kh}}^{\circ}$ and $\mathcal{F}_{\!\textit{Kh}}^{\lambda}$ with $\mathcal{E}^{\vee}$ . We provide a direct construction of both invariants.

Following [WebAlgebras] we call the web version of $H^{n}$ the Blanchet–Khovanov algebra. It is defined for any collection of oriented red and blue points $\Sigma$ that is balanced, i.e. bounds a web, as the direct sum

[TABLE]

where $\mathcal{B}$ is a cup basis of webs bounded by $\Sigma$ ; its elements play the role of cup diagrams for $H^{n}$ . Although $\mathfrak{W}^{\mathcal{B}}$ depends a priori on $\mathcal{B}$ , we show that different choices of basis lead to isomorphic algebras. Moreover, there is a special basis of webs—the red-over-blue basis—such that forgetting red facets in cup foams is compatible with multiplication. In particular, $\mathfrak{W}^{\mathcal{B}}$ admits a positive basis. This results immediately in an algebra isomorphism $\mathfrak{W}^{\mathcal{B}}\cong H^{n}\!$ , where $n$ is half of the blue points in $\Sigma$ . We further extend this construction to a 2-functor $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}\colon{\textbf{Foam}}^{\circ}\to{\textbf{Bimod}}$ following the construction of $\mathcal{F}_{\!\textit{Kh}}^{\circ}$ .

Suppose that $T$ is an oriented tangle, the input and output of which are balanced. Then all resolutions of $T$ are in ${\textbf{Foam}}^{\circ}$ and $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}$ can be applied to $\llbracket T\rrbracket_{\mathrm{F}}$ to produce a chain complex of bimodules $\mathit{C}_{\mathfrak{W}}(T)$ . We call it the Blanchet–Khovanov complex.

{PreTheoremC}

The 2-functor $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}$ is equivalent to $\mathcal{F}_{\!\textit{Kh}}^{\circ}\circ\mathcal{E}^{\vee}\!$ . In particular, the complexes $\mathit{C}_{\mathfrak{W}}(T)$ and $\mathit{CKh}(T)$ are isomorphic for any tangle $T\kern-0.5pt$ with balanced input and output.

The construction of a web version of Chen–Khovanov algebras is more challenging. We first describe two extensions of a sequence $\Sigma$ to a balanced one $\Sigma^{\circ}$ by inserting extra blue points to the left and to the right of $\Sigma$ . Then we pick a basis $\mathcal{B}$ of webs bounded by $\Sigma^{\circ}$ and the corresponding Blanchet–Khovanov algebra $\mathfrak{W}^{\mathcal{B}}$ . The extended Blanchet–Khovanov algebra $\mathfrak{A}^{\Sigma,\lambda}$ , where $\lambda\in\mathbb{Z}$ has the same parity as the number of blue points in $\Sigma$ , is a certain subquotient of $\mathfrak{W}^{\mathcal{B}}$ . Following the same procedure we associate a bimodule with a web and a bimodule map with a foam for every $\lambda\in\mathbb{Z}$ , obtaining a family of 2-functors $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\lambda}\colon{\textbf{Foam}}\to{\textbf{Bimod}}$ , each defined on the entire foam bicategory. As in the previous construction, $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\lambda}$ is compatible with relations between foams, so that applying it to $\llbracket T\rrbracket_{\mathrm{F}}$ results in an invariant chain complex of bimodules $\mathit{C}_{\mathfrak{W}}(T;\lambda)$ . We call it the extended Blanchet–Khovanov complex of $T$ .

We construct an explicit isomorphism $\mathfrak{A}^{\mathcal{B},\lambda}\cong A^{k,n-k}\!$ , where $n$ counts blue points in $\Sigma$ and $\lambda=n-2k$ . Contrary to the previous case, it is not enough to forget red facets in cup foams to get the isomorphism, because the basic webs from $\mathcal{B}$ may have too many blue arcs. This issue is resolved by stabilization—adding beneath webs and foams extra blue arcs and disks respectively. We then extend this isomorphism to bimodules and prove the following fact.

{PreTheoremD}

The 2-functor $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\lambda}$ is equivalent to $\mathcal{F}_{\!\textit{Kh}}^{\lambda}\circ\mathcal{E}^{\vee}\!$ . In particular, the complexes $\mathit{C}_{\mathfrak{W}}(T;\lambda)$ and $\mathit{CKh}(T;\lambda)$ are isomorphic for any tangle $T\kern-0.5pt$ .

All the isomorphisms are constructed explicitly and—in case nice bases are used—given by very simple formulas. Furthermore, by the discussion following Theorem B, the tangle homology computed with $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}$ and $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\lambda}$ are isomorphic to the Khovanov and Chen–Khovanov invariants respectively.

1.3 Functoriality of quantized annular Khovanov homology

The above results allow us to construct a strictly functorial version of the quantized annular Khovanov homology, which was the motivation for this paper. Combining Theorem LABEL:thm:FCKh-vs-Fweb with [QntHom, Proposition 6.6] we get

Corollary E.

Suppose $\Bbbk$ is flat over $\mathbb{Z}[q^{\pm 1}]$ . Then the quantum Hochschild homology groups $\mathit{qHH}_{\!i}(\mathfrak{A}^{\mathcal{B},\lambda})$ with coefficients in $\Bbbk$ vanish for $i>0$ , whereas the Chern character map

[TABLE]

is an isomorphism.

Choose now an oriented tangle $T$ that is bounded at both top and bottom by the same collection of oriented points $\Sigma$ . We define for its annular closure $\widehat{T}$ the quantum annular $\mathfrak{gl}_{2}$ complex as

[TABLE]

where $\mathcal{B}$ is a cup basis of webs bounded by $\Sigma$ and $C_{\mathfrak{A}}(T;\lambda)$ —the chain complex of bimodules obtained by applying $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\lambda}$ to $\llbracket T\rrbracket_{\mathrm{F}}$ . Corollary E together with [QntHom, Theorem B] imply the following.

Corollary F.

The quantum annular $\mathfrak{gl}_{2}$ homology $\mathit{Kh}_{\mkern-2.0mu\mathbb{A}_{q}}(L)$ is a triply graded invariant of annular links that is strictly functorial with respect to annular link cobordisms. Moreover, it admits an action of $\mathcal{U}_{q}(\mathfrak{gl}_{2})$ that commutes with the differential and the maps induced by annular link cobordisms.

It follows now from Theorem LABEL:thm:FCKh-vs-Fweb and the following discussion that $\mathit{Kh}_{\mkern-2.0mu\mathbb{A}_{q}}(L)$ is isomorphic with the quantized annular complex as constructed in [QntHom].

1.4 Further generalizations

The Khovanov homology has been extended by Asaeda, Przytycki, and Sikora to links in thickened surfaces [APS], but the functoriality has not been addressed until the resent paper of Quefellec and Wedrich [KhSurfaces]. There they have defined $\mathfrak{gl}_{2}$ foams in thickened oriented surfaces, and the natural question is whether the results of this paper can be extended to show equivalence of the two constructions. This is addressed in a follow up paper, where we also discuss foams in arbitrary 3-manifolds, including non-orientable ones.

Another natural question is about $\mathfrak{gl}_{N}$ foams for $N>2$ . Again there are two (bi)categories involved: of enhanced and not enhanced foams, the latter allowing only facets of labels up to $N-1$ . We expect that a proper generalization of this paper would prove equivalence of both (bi)categories, hence, also of the associated link homologies. Notice that functoriality of $\mathfrak{gl}_{N}$ homology has been shown in [KhRozFunctorial] using enhanced foams.

1.5 Organization of the paper

Section 2 provides a brief exposition of webs and foams. All the results presented there are well-known, except perhaps the choice of defining relations. Section 3 discusses shadings, their connection to webs and foams, and bicolored isotopies. It ends with a construction of a basis of the space of foams bounded by a given web. The equivalence of bicategories BN and Foam together with the local versions are constructed in Section 4, in which we also compare the two versions of the Khovanov bracket. Finally, Sections 5–7 provide detailed constructions of TQFT functors: a description of the tautological functor on ${\textbf{Foam}}(\emptyset)$ in terms of planar pictures, the constructions of the Blanchet–Khovanov algebras, their subquotients, and the 2-functors $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}$ and $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\lambda}\!$ .

1.6 Conventions and notation

Throughout the paper we fix a commutative unital ring $\Bbbk$ and linearity means $\Bbbk$ –linearity. We denote by $\{d\}$ the upward degree shift by $d$ , i.e. $M\{d\}_{i}=M_{i-d}$ for a graded module $M$ . Hence, a homogeneous $m\in M$ has degree $\deg(m)+d$ when seen as an element of $M\{d\}$ . We write $\mathit{Com}_{\!\raisebox{0.90417pt}{$ \scriptstyle/ $}\mkern-2.0muh}({\textbf{C}})$ for the homotopy category of a linear category C, the objects of which are formal complexes in C and morphisms—homotopy classes of chain maps.

Manifolds are assumed to be smooth (or at least piecewise smooth when necessary) and submanifolds are neat—that is $N\subset M$ is transverse to $\partial M$ and $\partial N=M\cap\partial M$ [DiffTop]. Orientation of a surface $S\subset\mathbb{R}^{3}$ is often identified with the canonical normal vector field $\nu$ , defined by the property that for each $p\in S$ the triple $(e_{1},e_{2},\nu_{p})$ , where $(e_{1},e_{2})$ is an oriented basis of $T_{p}S$ , is an oriented basis of $T_{p}\mathbb{R}^{3}$ . Such a vector field is unique up to an isotopy and can be found by the right hand rule.

1.7 Acknowledgments

The authors are grateful to the organizers of the program ,,Homology Theories in Low Dimensional Topology” in spring 2017 at the Isaac Newton Institute for Mathematical Sciences in Cambridge, where they have started to work on this project. A.B. and K.P. are supported by the NCCR SwissMAP founded by Swiss National Science Foundation.

2 Main players

This section provides basic definitions and facts about webs and foams. Most of the material is well-known [Blanchet, KhViaHowe], except perhaps the choice of defining relations, and the main purpose of this part is to fix notation and introduce terms used throughout the paper.

2.1 Webs

A web is an oriented trivalent graph with edges colored blue or red444 Red edges are drawn as double thick lines to make the difference visible when the paper is printed black and white.

in such a way, that at each vertex either two blue edges merge to a red one, or a red edge splits into two blue edges:

[TABLE]

In this paper webs will be always embedded in a disk $\mathbb{D}^{2}$ or a sphere $\mathbb{S}^{2}$ with a fixed basepoint $\ast$ that lies on $\partial\mathbb{D}^{2}$ in the case of a disk. Edges of a web in a disk can be attached transversely to the boundary circle away from $\ast$ ; each boundary point inherits then both the color and orientation from the attached edge: outwards (resp. inwards) oriented edges terminate with positive (resp. negative) points. A web is closed if its boundary is empty.

*Remark 2.1**.*

By moving the basepoint $\ast$ to the infinity, we can consider webs in $\mathbb{D}^{2}$ or $\mathbb{S}^{2}$ as embedded in a half plane $\mathbb{R}\times(-\infty,0]$ or a full plane $\mathbb{R}^{2}$ respectively.

Definition 2.2.

We write $\mathit{Web}$ for the module generated by isotopy555 Isotopies are assumed to fix points on the boundary circle.

classes of webs in a disk, modulo the local666 The word local means that two webs are identified if there is a disk outside of which the webs coincide and inside they look like in the pictures.

relations

[TABLE]

where the webs above can carry any coherent orientation unless indicated. For each collection of oriented red and blue points $\Sigma\subset\partial\mathbb{D}^{2}$ there is a submodule $\mathit{Web}(\Sigma)$ generated by webs bounded by $\Sigma$ and $\mathit{Web}$ is the direct sum of all of them.

Exercise 2.3.

Show that webs satisfy the following local relations:

[TABLE]

[Hint: Start with the left relation in (2.3).]**

Blue edges of a web $\omega$ form a crossingless tangle $\omega_{b}$ , which we call the underlying tangle of $\omega$ . In particular, it is a collection of disjoint circles when $\omega$ is closed. Write $\ell(\omega)$ for the number of blue loops in $\omega_{b}$ . Let $\mathrm{r}(\omega)$ be a web, the underlying tangle of which is $\omega_{b}$ with closed loops removed. We call it a reduction of $\omega$ . We construct it later using the bicolored isotopy argument and show the following fact, which implies in particular that $\mathrm{r}(\omega)$ does not depend on the placement of red edges.

Proposition 2.4.

Webs with same boundary and isotopic underlying tangles coincide in $\mathit{Web}$ . In particular, $\omega=(q+q^{-1})^{\ell(\omega)}\mathrm{r}(\omega)$ for any web $\omega$ .

Let $-\omega$ be the result of reversing orientation of all edges in a web $\omega$ . This operation preserves the relations (2.2) and (2.3), hence it induces an involution on $\mathit{Web}$ . It does not preserve the submodules $\mathit{Web}(\Sigma)$ , but there is a pairing

[TABLE]

which can be visualized by placing $-\omega$ and $\omega^{\prime}$ on the lower and upper hemisphere of a sphere and applying Proposition 2.4 to the resulting web (entirely red webs evaluate to $1$ ).

Lemma 2.5.

The pairing (2.5) is non-degenerate.

Proof.

Choose a nonzero $w\in\mathit{Web}(\Sigma)$ and write it as a linear combination $c_{1}\omega_{1}+\dots+c_{r}\omega_{r}$ of pair-wise non-isotopic webs $\omega_{1},\dots,\omega_{r}$ , the underlying tangles of which contain no loops. We may further assume that the polynomial $c_{1}$ contains a term $q^{d}$ with the maximal value of $|d|$ among all $c_{i}$ . Because $\ell(-\omega_{1}\cup\omega_{i})<\ell(-\omega_{1}\cup\omega_{1})$ for any $i\neq 1$ , the term $q^{d}(\omega_{1},\omega_{1})$ is not canceled in the expansion of $(w,\omega_{1})$ . Hence, $(w,\omega_{1})\neq 0$ . ∎

2.2 Foams

A foam is a collection of facets, oriented blue and red777 As in the case of webs, red facets of a foam are doubled in pictures.

surfaces, embedded in a 3-ball $B^{3}$ with boundary components attached transversely to $\partial B^{3}$ or glued together along singular curves called bindings in a way, such that locally two blue facets merge into a red one in an orientation preserving way as shown in Figure 1. Furthermore, blue facets may carry dots, but not the red ones, and bindings inherit orientation from blue facets. We say that a foam is closed if its boundary is empty. Otherwise it is bounded by a web in $\partial B^{3}$ . Notice that blue facets alone form a surface $S_{b}$ with dots, the underlying surface of $S$ . As in the case of webs, we fix a basepoint $\ast\in\partial B^{3}$ away from $\partial S$ . By moving it to infinity we can reinterpret foams as embedded in a half 3-space $\mathbb{R}^{2}\times(-\infty,0]$ .

There is a canonical cyclic order of facets attached to a binding that follows the right hand rule: point the thumb of your right hand along the binding curve and slightly bend the other fingers—they indicate the orientation of a small circle around the binding, hence, a cyclic order of facets. We call a blue facet positive or negative depending on whether it succeeds or precedes the red facet respectively. For non-embedded foams this cyclic order is usually provided explicitly by drawing small arrows around the binding, see [Blanchet].

Definition 2.6.

We write $\mathit{Foam}$ for the module generated by isotopy classes of foams in $B^{3}$ with the following local relations imposed:

•

sphere evaluations:

[TABLE]

•

neck cutting relations:

[TABLE]

•

dot reduction and dot moving relations:

[TABLE]

•

red facet detachments:

[TABLE]

Foams bounded by a web $\omega\subset\mathbb{S}^{2}$ (with $\ast\notin\omega$ ) generate a submodule $\mathit{Foam}(\omega)$ . As in the case of webs, $\mathit{Foam}$ is the direct sum of all these submodules.

*Remark 2.7**.*

The sign in (2.9) and (2.10) can be read easily from the direction of the canonical normal vector at the critical point on the red surface: it is positive exactly when the normal vector is directed towards the blue plane. For example, (2.9) can be written as

[TABLE]

When $\Bbbk$ is graded with $h$ and $t$ homogeneous in degree 2 and 4 respectively, then $\mathit{Foam}$ is a graded module with a foam $S$ being a homogeneous element in degree

[TABLE]

Here $\chi(S_{b})$ stands for the Euler characteristic of the underlying surface and $\mathit{dots}(S)$ counts dots carried by the foam.

The dot moving relation (the right one in (2.8)) takes a particularly simple form for $h=0$ : it allows to move a dot on the underlying surface at a cost of a sign. To have a similar interpretation in the general case, we introduce the dual dot as the difference

[TABLE]

The following exercise lists several relations satisfied by dual dots.

Exercise 2.8.

Show the following equalities between foams:

[TABLE]

The detaching relations (2.9) and (2.10) can take many other forms. For instance, redrawing them to make red facets horizontal results in

[TABLE]

Notice that in each case the sign can be read from the direction of the normal vector as explained in Remark 2.7.

We interpret the above relations later as isotopies between two surfaces, a blue and a red one. This will be a key ingredient in the proofs of the two facts listed below. In what follows we write $S\doteq S^{\prime}$ if foams $S$ and $S^{\prime}$ differ only by a sign and dualizing dots. For instance, $S\doteq S^{\prime}$ when $S^{\prime}$ is the result of moving a dot on the underlying surface of $S$ .

Proposition 2.9.

Let $S$ and $S^{\prime}$ be foams with isotopic underlying surfaces and same boundary. Then $S\doteq S^{\prime}$ in $\mathit{Foam}$ .

We prove the above proposition in the following section. An important consequence of it is the uniqueness (up to a sign) of a foam $\mathrm{cup}(\omega)$ , the underlying surface of which is a collection of disjoint disks bounded by $\omega_{b}$ . We call it the cup foam associated to $\omega$ . Then for any family $X$ of blue loops in $\omega_{b}$ we denote by $\mathrm{cup}(\omega,X)$ the cup foam with a dot placed on every blue disk that bounds a curve from $X$ . These foams constitute a linear basis of $\mathit{Foam}(\omega)$ as shown in Section 3.3.

Theorem 2.10.

Choose a closed web $\omega$ . The set $\{\mathrm{cup}(\omega,X)\ |\ X\subset BL(\omega)\}$ is a linear basis of $\mathit{Foam}(\omega)$ . In particular, $\mathit{Foam}(\omega)$ is a free graded module of rank $(q+q^{-1})^{\ell(\omega)}\!$ .

2.3 Decategorification

Fix a collection of red and blue oriented points $\Sigma\subset\mathbb{D}^{2}$ . A foam with corners in $\Sigma$ is a foam $S$ in $\mathbb{D}^{2}\times[0,1]$ with $S\cap\partial\mathbb{D}^{2}=\Sigma\times[0,1]$ . We gather them into a category ${\textbf{Foam}}(\Sigma)$ , in which

•

objects are webs bounded by $\Sigma$ with no relation imposed,

•

morphisms from $\omega_{0}$ to $\omega_{1}$ are generated by foams with corners in $\Sigma$ , with $\omega_{1}$ at the top and $-\omega_{0}$ at the bottom disk of $\mathbb{D}^{2}\times[0,1]$ , modulo the relations (2.6)–(2.10), and

•

the composition is given by stacking foams, one on top of the other.

We further enhance it to a graded additive category by introducing formal direct sums and formal degree shifts, so that objects are of the form $\omega_{1}\{d_{1}\}\oplus\dots\oplus\omega_{r}\{d_{r}\}$ , and redefining the degree of a foam $S\colon\omega_{0}\{a\}\to\omega_{1}\{b\}$ as

[TABLE]

where, as before, $\chi(S_{b})$ is the Euler characteristic of the underlying surface of $S$ and $\mathit{dots}(S)$ counts dots on $S$ , whereas $\#\Sigma_{b}$ is the number of blue points in $\Sigma$ . The reason for the last term is to make the identity foam a morphism of degree zero; it also makes the degree additive under the composition of foams. Furthermore, reinterpreting foams with corners as foams in $B^{3}$ leads to an isomorphism of graded $\Bbbk$ –modules

[TABLE]

for any webs $\omega$ and $\omega^{\prime}$ bounded by $\Sigma$ .

The orientation reversing diffeomorphism of the thickened disk $(p,t)\mapsto(p,1-t)$ induces a contravariant involutive functor

[TABLE]

that flips a foam vertically and reverses orientation of its facets. We check directly that all the defining relations (2.6)–(2.10) are preserved.

Foams with corners categorify webs. Indeed, web relations are lifted to isomorphisms:

[TABLE]

where the sign in the bottom left corner depends on the orientation of the edges. Therefore, there is a well-defined epimorphism $\gamma\colon\mathit{Web}(\Sigma)\to K_{0}({\textbf{Foam}}(\Sigma))\otimes_{\mathbb{Z}[q^{\pm 1}]}\Bbbk$ that takes a web $\omega$ to its class $[\omega]$ in the Grothendieck group.

Theorem 2.11.

The linear map $\gamma\colon\mathit{Web}(\Sigma)\to K_{0}({\textbf{Foam}}(\Sigma))\otimes_{\mathbb{Z}[q^{\pm 1}]}\Bbbk$ is an isomorphism.

Proof.

We have to show that $\gamma$ is injective. Consider a bilinear form $\langle-,-\rangle$ on $K_{0}({\textbf{Foam}}(\Sigma))$ defined for webs $\omega$ and $\omega^{\prime}$ as $\langle[\omega],[\omega^{\prime}]\rangle:=\operatorname{rk}_{q}\mathrm{Hom}_{{\textbf{Foam}}(\Sigma)}(\omega,\omega^{\prime})$ . It is well-defined, because the rank of the morphism space depends only on the images of webs in the Grothendieck group. Theorem 2.10 and the isomorphism (2.19) imply together that $\langle[\omega],[\omega^{\prime}]\rangle=(\omega,\omega^{\prime})$ , where the latter is the nondegenerate pairing from (2.5). Hence, $\gamma(w)=0$ forces $(w,-)=0$ , so that $w$ must be zero. ∎

2.4 Higher structures

It is common to consider webs embedded in a horizontal stripe $\mathbb{R}\times[0,1]$ instead of a disk. This is equivalent to picking two basepoints on $\partial\mathbb{D}^{2}$ , $\ast$ and $\ast^{\prime}$ , and placing them at the left and right infinities respectively. Such webs are morphisms of a linear category Web, the objects of which are finite collections of oriented red and blue points on a line, whereas the composition is defined by stacking stripes vertically:

[TABLE]

Formally, $\mathrm{Hom}_{{\textbf{Web}}}(\Sigma,\Sigma^{\prime})=\mathit{Web}(-\Sigma\cup\Sigma^{\prime})$ . This category is closely related to representations of $\mathcal{U}_{q}(\mathfrak{gl}_{2})$ [WebsHowe]: there is a monoidal functor $\mathcal{V}\colon{\textbf{Web}}\to{\textbf{Rep}}(\mathcal{U}_{q}(\mathfrak{gl}_{2}))$ such that

•

a blue positive (resp. negative) point is assigned the fundamental representation $V$ (resp. its dual $V^{\ast}$ ) and a red positive (resp. negative) point—the determinant representation $\mbox{\Large$ \wedge $}^{\mkern-6.0mu2}V$ (resp. $\mbox{\Large$ \wedge $}^{\mkern-6.0mu2}V^{\ast})$ , whereas a sequence of such points is assigned the tensor product of the corresponding representations,

•

the merge and split webs (2.1) are assigned the canonical inclusion and quotient maps between representations, and

•

cups and caps represent coevaluation and evaluation maps.

The relations between webs make the above functor faithful.

Define the weight of a point from $\Sigma$ according to the table below.

[TABLE]

The total weight $w(\Sigma)$ of $\Sigma$ is the sum of weights of its points. A quick analysis of the local model for webs (2.1) reveals that webs exist only between objects of the same weight. Hence, the category of webs decomposes into weight blocks ${\textbf{Web}}^{k}\kern-0.5pt$ , each spanned by objects of weight $k\in\mathbb{Z}$ . In particular, $\mathrm{Hom}_{{\textbf{Web}}}(\emptyset,\Sigma)\neq 0$ only when $w(\Sigma)=0$ ; such collections are called balanced.

In a similar matter one collects the foam categories ${\textbf{Foam}}(\Sigma)$ into a bicategory Foam, which also decomposes into blocks ${\textbf{Foam}}^{k}$ parametrized with $k\in\mathbb{Z}$ . Theorem 2.11 can be then rephrased to say that ${\textbf{Foam}}^{k}$ categorifies ${\textbf{Web}}^{k}$ , i.e. the category of webs is obtained by replacing morphism categories of Foam with their Grothendieck groups.

2.5 Blanchet evaluation formula

We end this section recalling the evaluation formula for closed foams in a 3-ball $B^{3}$ following [Blanchet]. It requires two 2-dimensional TQFTs, one for blue and one for red facets. Each is uniquely determined by the (associative) commutative Frobenius algebra assigned to a circle. We choose the algebras

[TABLE]

for blue and red circles respectively, where $h,t\in\Bbbk$ are fixed parameters (the standard choice is $h=t=0$ ). The comultiplications and counits are defined by the formulas

[TABLE]

A dot on a blue surface is interpreted as the multiplication with $X$ . Notice that $h-X$ , which represents a dual dot, satisfies the polynomial relation defining $A_{b}$ , so that $\overline{X}:=h-X$ extends to a conjugation compatible with multiplication. One checks directly that $\overline{\Delta_{b}(a)}=-\Delta_{b}(\overline{a})$ and $\epsilon_{b}(\overline{a})=-\epsilon_{b}(a)$ for any $a\in A_{b}$ .

When $\Bbbk$ is graded with $h$ and $t$ homogeneous in degree 2 and 4 respectively, then we make $A_{b}$ a graded algebra by setting $\deg(X)=2$ ; comultiplication and counit increase and decrease the degree by $2$ respectively. Assigning now $A_{b}\{-1\}$ to a blue circle produces a graded TQFT: $\deg(1)=-1$ and $\deg(X)=+1$ , in which case both multiplication and comultiplication are homogeneous in degree 1, matching the degree of a saddle. Likewise for the unit and counit. The other TQFT is upgraded by inheriting the grading on $A_{r}$ from $\Bbbk$ .

Assume that a closed foam $S$ is obtained from a blue surface $S_{b}$ and a red one $S_{r}$ by identifying boundary circles $C^{+}_{i},C^{-}_{i}\subset\partial S_{b}$ with $C^{0}_{i}\subset\partial S_{r}$ for $1\leq i\leq m$ , such that $C^{+}_{i}$ and $C^{-}_{i}$ come from the positive and negative facet respectively. Let

[TABLE]

be the elements assigned by the two TQFTs to the blue and red surface, where the first factor in $A_{b}\otimes A_{b}$ corresponds to $C_{i}^{+}$ and the second to $C_{i}^{-}$ . The evaluation assigns to $S$ the value

[TABLE]

where $\pi\colon A_{b}\otimes A_{b}\to A_{b}$ sends $x\otimes y$ to $x\overline{y}$ and $\eta\colon A_{r}\to A_{b}$ is the inclusion of algebras; the trace map $\operatorname{tr}\colon A_{b}\otimes A_{b}\to\Bbbk$ is the composition of the multiplication with the counit of $A_{b}$ .

Example 2.12.

Let $S$ be a blue sphere with a red disk inside and one dot, as shown below. It decomposes into three cups, two blue and a red one, where one of the blue cups carries a dot:

[TABLE]

The orientation of the binding determines that the dotted cup is attached to the negative boundary. Hence,

[TABLE]

resulting in $Z(S)=\operatorname{tr}(1\otimes\overline{X})=\epsilon_{b}(h-X)=-1$ .

The relations (2.15) and (2.6) evaluate the foam $S$ from the example above to $-1$ as well. This is not a coincidence: the defining relations were looked up in the kernel of $Z$ . In fact, Proposition 2.9 implies a stronger statement. It was first proven in [Blanchet].

Theorem 2.13 (cp. [Blanchet]).

The evaluation (2.23) descends to an isomorphism $Z\colon\mathit{Foam}(\emptyset)\to\Bbbk$ .

Proof.

We first check that $Z$ is well-defined, i.e. it preserves the relations (2.6)–(2.10). Those involving facets of one color can be checked directly, whereas moving a dot through an $i$ -th binding corresponds to taking it from a facet attached to $C_{i}^{+}$ (multiplication by $X$ ) and placing it on the facet attached to $C_{i}^{-}$ (multiplication by $\overline{X}=h-X$ ). Hence, (2.8) is satisfied. We follow now Example 2.12 to compute

[TABLE]

which immediately implies (2.9): using (2.7) cut both the red cylinder and the plane around the binding to obtain a sum of three foams, each consisting of a red cup, a blue plane, and a blue sphere with a red membrane inside. Two of these foams have an additional dot, one on the plane and the other the sphere; only the latter term survives and the sign comes from (2.24). We leave (2.10) as an exercise.

Assume now that $Z(S)=0$ . By Proposition 2.9, $S$ coincides up to a sign with an entirely blue foam $S^{\prime}$ , which is the blue surface $S_{b}$ , perhaps with some dots replaced with dual dots. However, applying the blue neck cutting relation (2.7) to any component of positive genus reduces $S^{\prime}$ further to a sum of collections of dotted spheres. These in turn can be completely evaluated with (2.8) and (2.6). Hence, $S^{\prime}=Z(S^{\prime})=0$ , which shows that $Z$ is invertible. ∎

3 Shadings and a basis of foams

This part is the backbone of the paper. We introduce here shadings of manifolds, use them to construct webs and foams, and prove the bicolored isotopy lemma: isotopic shadings encode equal webs and foams (the latter up to a sign and type of dots). Using this language we introduce then a basis of foams that is especially easy to visualize.

3.1 Shadings and trivalent manifolds

A shading of a manifold $M^{n}$ consists of two codimension 1 submanifolds, an oriented $U_{r}$ and a non-oriented $U_{b}$ , that are transverse to each other and to $\partial M^{n}$ , together with a checkerboard coloring of $M^{n}$ : a choice of color, white or black, for each connected component of the complement of $U_{r}\cup U_{b}$ . We refer to $U_{r}$ and $U_{b}$ as red and blue respectively. The components of the intersectoin $U_{r}\cap U_{b}$ are called bindings; they decompose both $U_{r}$ and $U_{b}$ into facets. Finally, we refer to the components of the complement of $U_{r}\cup U_{b}$ in $M^{n}$ as regions.

Lemma 3.1.

Assume $M^{n}$ is simply connected and fix a point $\ast\in M^{n}\!$ . Then a pair of codimension 1 submanifolds of $M^{n}\!$ , that are transverse to each other and away from $\ast$ , determines a unique shading of $M^{n}\!$ with the region containing $\ast$ painted white.

Proof.

Given a pair of transverse codimension 1 submanifolds $(U_{r},U_{b})$ we construct a desired shading as follows. Given $p\in M^{n}\setminus(U_{r}\cup U_{b})$ choose a path $\gamma$ from $\ast$ to $p$ , transverse to both $U_{r}$ and $U_{b}$ , and let $d(\gamma):=\#(\gamma\cap U_{r})+\#(\gamma\cap U_{b})$ count the intersection points of $\gamma$ with both submanifolds. Color $p$ white or black depending on whether $d(\gamma)$ is even or odd. Because $M^{n}$ is simply connected, the parity of $d(\gamma)$ does not depend on the choice of $\gamma$ and the color of $p$ is well-defined. ∎

*Remark 3.2**.*

It follows from Lemma 3.1 that every codimension 1 submanifold $U$ of a simple connected manifold $M^{n}$ admits a standard orientation: the one induced from white regions, when $U$ is considered as a shading with $U_{r}=\emptyset$ . When $M^{n}$ is a line and $U$ a collection of blue points, then the standard orientation on $U$ is the alternating one. Likewise for the case $M^{n}=\mathbb{S}^{1}\!$ , assuming the cardinality of $U$ is even (otherwise it does not extend to a shading).

A trivalent manifold embedded in $M^{n}$ is a generalization of webs and foams. It is a collection of facets, oriented codimension 1 submanifolds colored blue or red, with boundary components attached transversely to $\partial M^{n}$ or glued together along bindings in a way, such that locally two blue facets merge into a red one. In other words, each point of a trivalent manifold has a neighborhood diffeomorphic to either $\mathbb{R}^{n-1}$ or $Y\times\mathbb{R}^{n-2}$ , where $Y$ is an oriented merge or a split from (2.1).

Given a shading $(U_{r},U_{b})$ of $M^{n}$ we construct a trivalent manifold $\Gamma(U_{r},U_{b})$ by examining the orientation on facets induced from white regions:

•

blue facets inherit the orientation,

•

red facets are preserved (,,amplified”) if the induced orientation agrees with the given one or annihilated otherwise.

An example is presented in Figure 2.

In particular, $\Gamma(\emptyset,U_{b})$ is $U_{b}$ with its standard orientation as defined in Remark 3.2. It appears that every trivalent manifold arises this way, the proof of which is presented below and visualized in Figure 3. Hence, shadings can be considered as completions of trivalent manifolds, because of which we shall refer to shadings of $\mathbb{D}^{2}$ and $B^{3}$ as completed webs and completed foams respectively.

Lemma 3.3.

Choose a trivalent manifold $V\subset M^{n}\!$ , such that $\partial V=\Gamma(\tilde{U}_{r},\tilde{U}_{b})$ for some shading $(\tilde{U}_{r},\tilde{U}_{b})$ of $\partial M^{n}\!$ . Then there exists a shading $(U_{r},U_{b})$ of $M^{n}\kern-0.5pt$ that restricts to $(\tilde{U}_{r},\tilde{U}_{b})$ on $\partial M^{n}\!$ and satisfies $\Gamma(U_{r},U_{b})=V\kern-0.5pt$ .

Proof.

Consider the orientation of $\partial M^{n}$ induced from $M^{n}$ and reverse it at all points painted black in the given shading. Then the boundary of any region $R\subset M^{n}$ is a union of facets of $V$ and regions in $\partial M^{n}\!$ , such that oppositely oriented components meet only in two situations: when they are both contained in the boundary (so that they meet at a facet of $\partial V$ ) or both are blue facets of $V$ adjacent to a red facet outside of $R$ . Consider the union of those components of $\partial R$ , the orientation of which does not match the one induced from $R$ . They constitute certain oriented $(n-1)$ -dimensional submanifolds $U_{1},\dots,U_{k}$ . Taking a red colored copy $U^{\prime}_{i}$ of each $U_{i}$ push its interior inside $R$ and paint the newly created region black. Repeating this for each region produces a desired shading. ∎

A useful consequence of Lemma 3.3 is that tangles and surfaces can be extended to webs and foams with given boundary. Recall that a collection $\Sigma\subset\partial\mathbb{D}^{2}$ of oriented red and blue points is balanced if it bounds a web, which is equivalent to being of weight zero.

Proposition 3.4.

Let $\Sigma\subset\partial\mathbb{D}^{2}$ be a balanced collection of oriented red and blue points and $\tau$ a tangle bounded by $\Sigma_{b}$ . Then there exists a web $\omega$ bounded by $\Sigma$ with $\omega_{b}=\tau\kern-0.5pt$ . 2. 2)

Let $\omega\subset\partial B^{3}$ be a web and $W\kern-0.5pt$ a surface bounded by $\omega_{b}$ . Then there is a foam $S$ bounded by $\omega$ with $S_{b}=W\kern-0.5pt$ .

Proof.

Extend $\Sigma$ to a shading $\widetilde{\Sigma}=(\Sigma_{r}\cup\Sigma^{\prime}_{r},\Sigma_{b})$ . Then $\widetilde{\Sigma}$ has an even number of points and the orientation of points from $\Sigma$ matches the one induced from white regions. Let $b$ , $r$ , and $r^{\prime}$ be the sums of orientations of blue points in $\Sigma$ , red points in $\Sigma$ , and red points added to $\widetilde{\Sigma}$ respectively. Then $b+2r=0$ , because $\Sigma$ is balanced, and $b+r+r^{\prime}=0$ , because the orientation of points in $\widetilde{\Sigma}$ alternate. Subtracting the two equalities reveals that $r-r^{\prime}=0$ . It follows that there is an oriented collection of disjoint intervals $\tau_{r}\subset\mathbb{D}^{2}$ bounded by $\widetilde{\Sigma}_{r}$ , the orientation of which agree with the points from $\Sigma_{r}$ and disagree with those from $\Sigma^{\prime}_{r}$ . Hence, $\omega:=\Gamma(\tau_{r},\tau)$ is the desired web.

The second statement is even easier to show. Extend the web $\omega$ to a shading $\alpha$ . Then $\alpha_{r}$ is a collection of disjoint loops and each such collection bounds a family $W_{r}$ of disjoint disks in $B^{3}$ . Therefore, $S:=\Gamma(W_{r},W)$ is the desired foam. ∎

3.2 Bicolored isotopies

Choose an isotopy $\Phi$ of $M^{n}$ and a subset $A$ . The set $\operatorname{Tr}_{\Phi}(A)=\{(\Phi_{t}(a),t)\ |\ a\in A,t\in[0,1]\}$ is called the trace of $A\subset M^{n}$ under $\Phi$ [DiffTop]. We say that a pair of isotopies $(\Phi,\Psi)$ of $M^{n}$ is an isotopy of a shading $(U_{r},U_{b})$ if $(\operatorname{Tr}_{\Phi}(U_{r}),\operatorname{Tr}_{\Phi}(U_{b}))$ is a shading of $M^{n}\times[0,1]$ that coincides with $(U_{r},U_{b})$ at the level $t=0$ . When $M^{n}$ is a disk, then a generic pair of isotopies can be encoded by a sequence of bigon moves

[TABLE]

whereas in case of a 3-ball two moves are necessary:

[TABLE]

In each move a shading of one side determines a shading of the other. Hence, we obtain the following characterization of isotopies of shadings in these cases.888 This can be extended to all manifolds by a detailed analysis of singular levels of a pair of isotopies.

Lemma 3.5.

*Two shadings of $\mathbb{D}^{2}$ or $B^{3}$ are isotopic if and only if their codimension 1 components are isotopic, possibly by different isotopies. *

When a basepoint $\ast\in M^{n}$ is present, then one must be careful how it behaves under the isotopy. There is no problem when $\Psi$ and $\Phi$ coincide at $\ast$ (and in this paper we always assume that both $\Psi$ and $\Phi$ fix $\ast$ ). Otherwise, the basepoint should stay at the same region if possible. However, when the region disappears, then the basepoint has to reappear in a white region. For instance, when $\ast$ lies in the small bigon on the left hand side of (3.1), then it reappears between the two strands on the right hand side. The same can be done for both moves in (3.2).

Recall from Section 2.2 that we write $S\doteq S^{\prime}$ for foams $S$ and $S^{\prime}$ if they agree up to a sign and replacing some dots with their duals.

Lemma 3.6 (Bicolored Isotopy).

$\Gamma(\alpha_{r},\alpha_{b})=\Gamma(\alpha^{\prime}_{r},\alpha^{\prime}_{b})$ * in $\mathit{Web}$ if $(\alpha_{r},\alpha_{b})$ and $(\alpha^{\prime}_{r},\alpha^{\prime}_{b})$ are isotopic shadings of $\mathbb{D}^{2}\!$ .* 2. 2)

$\Gamma(W_{r},W_{b})\doteq\Gamma(W^{\prime}_{r},W^{\prime}_{b})$ * in $\mathit{Foam}$ if $(W_{r},W_{b})$ and $(W^{\prime}_{r},W_{b}^{\prime})$ are isotopic shadings of $B^{3}\!$ .*

Proof.

It is enough to consider the case of elementary isotopies. When applied to each side of the bigon move (3.1), $\Gamma$ removes red edges in both pictures from the same side of the blue line. Hence, $\Gamma(\alpha_{r},\alpha_{b})$ and $\Gamma(\alpha^{\prime}_{r},\alpha^{\prime}_{b})$ are related by the left relation in either (2.3) or (2.4). Likewise, the moves (3.2) correspond to the detaching relations (2.9) and (2.10). ∎

The above result has far reaching consequences when paired with Lemma 3.3. The statements about comparing webs and foams with isotopic blue pieces follows, which in turn were used in the proof of Theorem 2.13 to show bijectivity of the Blanchet evaluation map $Z$ .

Proof of Proposition 2.4.

Let $\omega$ and $\omega^{\prime}$ have isotopic underlying tangles and take the trace of $\omega_{b}$ under this isotopy as $S_{b}$ ; it is the underlying surface of a foam $S\colon\omega\to\omega^{\prime}$ due to Proposition 3.4. Extend the foam to a shading $(\widetilde{S}_{r},S_{b})$ of $\mathbb{D}^{2}\times[0,1]$ . When in generic position, it can be represented by a finite sequence of level sets, such that in between any two consecutive levels $\widetilde{S}_{r}$ has either no critical points (so that the level sets are related by the Bicolored Isotopy Lemma) or a unique Morse type critical point—a cap, a cup, or a saddle—in which case the corresponding webs coincide (if the affected red edges are erased) or are identified by the right relations in (2.2) and (2.3) (if the red edges survive). Notice that $S_{b}$ has no critical points.

For the second part, extend $\omega$ to a shading $(\widetilde{\omega}_{r},\omega_{b})$ of $\mathbb{D}^{2}$ and isotope closed blue loops, so that they do not intersect $\omega_{r}$ . Applying $\Gamma$ results in a new web $\omega^{\prime}$ that coincides with $\omega$ as shown above. Removing blue circles from $\omega^{\prime}$ results in $\mathrm{r}(\omega)$ and the desired equality follows from (2.2). ∎

Proof of Proposition 2.9.

Let foams $S_{1}$ and $S_{2}$ have isotopic blue parts. Extend them to shadings $W_{1}$ and $W_{2}$ respectively and pick a ball $\mathcal{O}$ in the interior of $B^{3}$ , outside of which the red facets of the shadings coincide. Using Lemma 3.6 isotope blue facets away from $\mathcal{O}$ (this may dualize dots), reducing the problem to showing equality for foams with only red facets. In such case, use the neck cutting relation (2.7) to reduce each foam to a collection of disjoint disks and spheres; this may change the sign of the foam. The thesis follows, because each red sphere evaluates to $-1$ and the disks are uniquely determined up to an isotopy by the boundary circles. ∎

It follows immediately from Proposition 2.9 that the foam used in the proof of Proposition 2.4 is invertible. That would be enough to prove the latter if we knew that Foam categorifies $\mathit{Web}$ . However, the proof of the categorification result is based on Theorem 2.10, which is proven only in the next section.

3.3 Cup foams

We will now apply the above results to show that cup foams, as defined in Section 2.2, constitute a free basis of spaces of foams. In particular, the category of foams is non-degenerate.

Let $\omega$ be a closed web, so that $\omega_{b}$ is a collection of blue loops. Orient them in a standard way (see Remark 3.2) and pick a foam $I_{\omega}\in\mathrm{Hom}_{{\textbf{Foam}}(\emptyset)}(\omega_{b},\omega)$ with $\omega_{b}\times[0,1]$ as its underlying surface; the existence of such a foam follows from Proposition 3.4. According to Proposition 2.9, there is a sign $\mathrm{sgn}\mskip 3.0mu(\omega)=\pm 1$ satisfying

[TABLE]

where $\mathrlap{\phantom{I}{}^{!}}I_{\omega}\in\mathrm{Hom}_{{\textbf{Foam}}}(\omega,\omega_{b})$ is the vertical flip of $I_{\omega}$ as defined in (2.20). The sign can be also computed directly as

[TABLE]

where $C_{\omega}\in\mathrm{Hom}_{{\textbf{Foam}}}(\emptyset,\omega_{b})$ is a collection of disks bounded by $\omega_{b}$ and $\mathrlap{\phantom{C}{}^{\bullet}}C_{\omega}\in\mathrm{Hom}_{{\textbf{Foam}}}(\omega_{b},\emptyset)$ the same collection, except that each disk is decorated by a dot. Hence, $\mathrm{sgn}\mskip 3.0mu(\omega)$ is a well-defined integer, which we call the sign of the web $\omega$ .

Lemma 3.7.

The sign $\mathrm{sgn}\mskip 3.0mu(\omega)$ does not depend on the choice of $I_{\omega}$ .

Proof.

Let $S\in\mathrm{Hom}_{{\textbf{Foam}}}(\omega_{b},\omega)$ be another foam with $S_{b}=\omega_{b}\times[0,1]$ . Then $S=\pm I_{\omega}$ by Proposition 2.9 and $S^{!}\,S=\mathrlap{\phantom{I}{}^{!}}I_{\omega}\,I_{\omega}$ , because the same sign relates $S^{!}$ with $\mathrlap{\phantom{I}{}^{!}}I_{\omega}$ . ∎

Let $BL(\omega)$ be the collection of blue loops in $\omega$ . For each subset $X\subset BL(\omega)$ we construct the cup foam $\mathrm{cup}(\omega,X)$ by attaching blue disks to the input of $I_{\omega}$ and placing a dot on each disk bounded by a loop from $X$ . Notice that red facets of $\mathrm{cup}(\omega,X)$ are above all dots and minima of blue facets. Therefore, we say that $\mathrm{cup}(\omega,X)$ is a red-over-blue cup foam decorated by $X\kern-0.5pt$ . We construct likewise a cap foam $\mathrm{cap}(\omega,X)\in\mathrm{Hom}_{{\textbf{Foam}}(\emptyset)}(\omega,\emptyset)$ by reflecting $\mathrm{cup}(\omega,X)$ vertically and replacing each dot with the dual one scaled by $-1$ . For instance, we have the following correspondence between cup and cap foams bounded by two blue loops:

[TABLE]

Let us now represent a foam $S\in\mathrm{Hom}_{{\textbf{Foam}}(\emptyset)}(\omega,\omega^{\prime})$ by a vertical cylinder labeled $S\kern-0.5pt$ , with $\omega$ and $\omega^{\prime}$ at the bottom and top disk respectively. When no label is present, it is understood that $\omega=\omega^{\prime}$ and the cylinder represents the identity foam $\omega\times[0,1]$ . We emphasize the cases $\omega=\emptyset$ and $\omega^{\prime}=\emptyset$ by drawing a cup or a cap instead and, to simplify notation, we decorate it directly with $X\subset BL(\omega)$ when $S$ is a cup or a cap foam:

[TABLE]

Moreover, $X^{c}:=BL(\omega)\setminus X$ stands for the complement of a subset $X\subset BL(\omega)$ .

Lemma 3.8.

Foams satisfy the following relations:

[TABLE]

Proof.

From the construction of cup and cap foams

[TABLE]

and the right hand side is a collection of spheres, each carrying at most one regular and one dual dot, scaled by $(-1)^{|Y|}$ . Such a sphere evaluates to 1 or $-1$ when it carries either one regular or one dual dot respectively and vanishes otherwise (see Exercise 2.8). Hence, (3.3) follows.

The second relation follows from the equality $\omega\times[0,1]=\mathrm{sgn}\mskip 3.0mu(\omega)\,I_{\omega}\,\mathrlap{\phantom{I}{}^{!}}I_{\omega}$ and the neck cutting relation from Exercise 2.8. ∎

We are ready to prove that cup foams form a linear basis of foams.

Proof of Theorem 2.10.

The first relation of Lemma 3.8 implies that cup foams are linearly independent. To show that they generate $\mathit{Foam}(\omega)\cong\mathrm{Hom}_{{\textbf{Foam}}(\emptyset)}(\emptyset,\omega)$ , use the second relation to write a foam $S$ bounded by $\omega$ as a sum

[TABLE]

which is a linear combination of cup foams, because closed foams evaluate to scalars. Finally,

[TABLE]

as the underlying surface of the cup foam consists of $\ell$ disks decorated by $|X|$ dots, so that

[TABLE]

as desired. ∎

4 Equivalences of foam and cobordism categories

In this section we prove Theorems A and B, which state that foams and Bar-Natan cobordisms constitute equivalent (bi)categories. We then relate the formal complexes $\llbracket T\rrbracket$ and $\llbracket T\rrbracket_{\mathrm{F}}$ associated with a tangle $T\kern-0.5pt$ .

4.1 Embedding cobordisms into foams

Fix a balanced collection $\Sigma\subset\partial\mathbb{D}^{2}$ away from a fixed basepoint $\ast\in\partial\mathbb{D}^{2}$ and write $\Sigma_{b}$ for the subset consisting of all blue points from $\Sigma$ . Consider first the case when $\Sigma=\Sigma_{b}$ and the points are oriented in a standard way as explained in Remark 3.2. This means that, when following the orientation of the boundary circle, the first point after the basepoint is negative and then the orientation alternates. Theorem B is in this case a direct consequence of Proposition 2.4 and Theorem 2.10: each web is isomorphic to an entirely blue one (and each such web is a flat tangle equipped with the standard orientation) and for such webs $\omega$ and $\omega^{\prime}$ the cup basis of $\mathrm{Hom}_{{\textbf{Foam}}(\Sigma)}(\omega,\omega^{\prime})$ consists of foams with no red facets. Hence, the naive map $\mathrm{Hom}_{{\textbf{BN}}(\Sigma)}(\omega,\omega^{\prime})\to\mathrm{Hom}_{{\textbf{Foam}}(\Sigma)}(\omega,\omega^{\prime})$ that orients a cobordism in a standard way does the job. A little more work has to be done to cover the general case.

Lemma 4.1.

There is a web $E_{\Sigma}\subset\mathbb{S}^{1}\times[0,1]$ bounded by $\Sigma$ at $\mathbb{S}^{1}\times\{1\}$ and standardly oriented $\Sigma_{b}$ at $\mathbb{S}^{1}\times\{0\}$ , which is disjoint from $\{\ast\}\times[0,1]$ and with $\Sigma_{b}\times[0,1]$ as the underlying tangle.

Proof.

Let $\tau$ be a collection of radial blue intervals connecting blue points at $\mathbb{S}^{1}\times\{0\}$ with those at $\mathbb{S}^{1}\times\{1\}$ . Cut the annulus to a disk along $\{\ast\}\times[0,1]$ and apply Proposition 3.4 to get a desired web. ∎

*Remark 4.2**.*

The extension of a tangle to a web is constructed in Lemma 3.3 from a shading of the disk, which is by no means unique. In case of an annulus, however, the situation is different: there is a unique up to an isotopy family of counter-clockwise oriented arcs that bounds a given collection of oriented points at the outer boundary circle. Some of the arc may intersect the interval $\{\ast\}\times[0,1]$ ; moving them through the hole results in a preferred shading and a preferred web $E_{\Sigma}$ .

Inserting a tangle inside the web $E_{\Sigma}$ and a surface inside the foam $E_{\Sigma}\times[0,1]$ results in a functor $\mathcal{E}_{\Sigma}\colon{\textbf{BN}}(\Sigma_{b})\to{\textbf{Foam}}(\Sigma)$ , as it preserves units and composition.

Theorem B.

The functor $\mathcal{E}_{\Sigma}\colon{\textbf{BN}}(\Sigma_{b})\to{\textbf{Foam}}(\Sigma)$ is an equivalence of categories.

Proof.

It follows from Propositions 2.4 and 2.9 that $\mathcal{E}_{\Sigma}$ is essentially surjective and full. Faithfulness follows from Theorem 2.10: both $\mathrm{Hom}_{{\textbf{BN}}(\Sigma_{b})}(\omega_{b},\omega_{b}^{\prime})$ and $\mathrm{Hom}_{{\textbf{Foam}}(\Sigma)}(\omega,\omega^{\prime})$ are free graded modules of graded rank $(q+q^{-1})^{\ell}\!$ , where $\ell$ counts blue loops in $-\omega\cup\omega^{\prime}\!$ . ∎

4.2 A coherent way to forget red facets

The inverse functor to $\mathcal{E}_{\Sigma}$ forgets red facets of foams, but it may also change the sign. To construct it explicitly, fix for each web $\omega$ an invertible foam $I_{\omega}\in\mathrm{Hom}_{{\textbf{Foam}}(\Sigma)}(\mathcal{E}_{\Sigma}(\omega_{b}),\omega)$ . Given a non-vanishing foam $S\colon\omega\to\omega^{\prime}$ consider the square

[TABLE]

where $\mathrm{sgn}\mskip 3.0mu(S)=\pm 1$ is the unique sign for which the square commutes, i.e.

[TABLE]

It exists by Proposition 2.9. Both sides are foams bounded by the web $-\mathcal{E}_{\Sigma}(\omega_{b})\cup\omega^{\prime}$ and they evaluate to a nonzero scalar after glued with some cup foam $C$ .999 Explicitly, $C=\mathrm{cup}(-\mathcal{E}_{\Sigma}(\omega_{b})\cup\omega^{\prime},X)$ where $X$ contains exactly one boundary circle of each genus 0 component of $S_{b}$ that does not carry a dot.

Therefore,

[TABLE]

where $Z$ is the Blanchet evaluation map.

Proposition 4.3.

The assignment

[TABLE]

defines a functor $\mathcal{E}^{\vee}_{\Sigma}\colon{\textbf{Foam}}(\Sigma)\to{\textbf{BN}}(\Sigma_{b})$ inverse to $\mathcal{E}_{\Sigma}$ .

Proof.

We have to check that the sign $\mathrm{sgn}\mskip 3.0mu(S)$ is multiplicative with respect to composition of foams. For that pick foams $S^{\prime}\in\mathrm{Hom}_{{\textbf{Foam}}(\Sigma)}(\omega,\omega^{\prime})$ and $S^{\prime\prime}\in\mathrm{Hom}_{{\textbf{Foam}}(\Sigma)}(\omega^{\prime},\omega^{\prime\prime})$ , such that the composition $S^{\prime\prime}S^{\prime}$ does not vanish. Then

[TABLE]

which forces $\mathrm{sgn}\mskip 3.0mu(S^{\prime\prime}S^{\prime})=\mathrm{sgn}\mskip 3.0mu(S^{\prime\prime})\mathrm{sgn}\mskip 3.0mu(S^{\prime})$ . To end the proof, we check directly that $\mathcal{E}^{\vee}_{\Sigma}\circ\mathcal{E}_{\Sigma}$ is the identity functor on ${\textbf{BN}}(\Sigma_{b})$ , whereas the collection of the invertible foams $I_{\omega}$ constitute a natural isomorphism between $\mathcal{E}_{\Sigma}\circ\mathcal{E}^{\vee}_{\Sigma}$ and the identity functor on ${\textbf{Foam}}(\Sigma)$ . ∎

Example 4.4.

Let $\omega$ be a blue circle oriented clockwise. This is the orientation induced from the unbounded region, hence standard, so that $\omega=\omega_{b}$ and $\mathcal{E}^{\vee}_{\emptyset}$ simply forgets orientation:

[TABLE]

However, when $\omega$ is oriented counter-clockwise, then the invertible foam $I_{\omega}$ is a cylinder with a red membrane and removing the membrane may cost a sign:

[TABLE]

*Remark 4.5**.*

Although the construction of $\mathcal{E}^{\vee}_{\Sigma}$ depends on the choice of foams $I_{\omega}$ , the functor is unique up to a unique natural isomorphism. To see this directly, suppose that $\tilde{\mathcal{E}}^{\vee}_{\Sigma}$ is constructed using a different family of foams $\tilde{I}_{\omega}$ . Then $\tilde{I}_{\omega}=s(\omega)I_{\omega}$ for a well-defined sign $s(\omega)=\pm 1$ and it follows from a direct computation that the collection of morphisms $\iota_{\omega}:=s(\omega)\cdot\omega_{b}\times[0,1]$ is a natural isomorphism from $\mathcal{E}^{\vee}_{\Sigma}$ to $\tilde{\mathcal{E}}^{\vee}_{\Sigma}$ .

4.3 An equivalence of bicategories

Recall that a 1-morphism $f\colon x\to y$ in a bicategory C is an equivalence if there exists $g\colon y\to x$ such that the compositions $f\circ g$ and $g\circ f$ are isomorphic to identity 1-morphisms. A 2-functor $\mathcal{F}\colon{\textbf{C}}\to{\textbf{D}}$ is an equivalence of bicategories when

•

it is a local equivalence, that is the functor $\mathcal{F}_{x,y}\colon{\textbf{C}}(x,y)\to{\textbf{D}}(\mathcal{F}(x),\mathcal{F}(y))$ is an equivalence of categories for all objects $x,y$ of C, and

•

it is essentially surjective: each object of D is equivalent to an object of the form $\mathcal{F}(x)$ .

Indeed, the above conditions imply the existence of an inverse of $\mathcal{F}$ [BasicBicats].

There is a 2-functor

[TABLE]

that equips points, tangles, and cobordisms with the standard orientation.101010 Recall the convention that the basepoint $\ast$ is placed at the left infinity, so that the left unbounded region is painted white. This implies in particular that the left most point of an object of BN receives the positive orientation and the left most vertical strand of a 1-morphism is oriented upwards.

It is a local equivalence due to Theorem B, but not essentially surjective: objects from the image of $\mathcal{E}^{0}$ have weight 0 or 1, so that the whole image is contained in ${\textbf{Foam}}^{0}\sqcup{\textbf{Foam}}^{1}$ . We fix this by enlarging the source bicategory to ${\textbf{wBN}}:={\textbf{BN}}\times\mathbb{Z}$ , the product of BN with $\mathbb{Z}$ seen as a discrete bicategory. In other words, objects of wBN are pairs $(\Sigma,k)$ consisting of an object $\Sigma$ from BN and a number $k\in\mathbb{Z}$ , whereas morphism categories are copied directly from BN:

[TABLE]

We then extend (4.2) to a 2-functor

[TABLE]

in a way, such that $(\Sigma,k)$ is taken to the collection $\mathcal{E}^{0}(\Sigma)$ with $|k|$ red points added to the right, all positive when $k>0$ and negative otherwise. Likewise for 1- and 2-morphisms: $\mathcal{E}$ takes a tangle $\tau$ (resp. a cobordism $W$ ), orients it in a standard way, and adds to the right $|k|$ vertical red lines (resp. vertical red squares) with the appropriate orientation.

Theorem A.

The 2-functor $\mathcal{E}\colon{\textbf{wBN}}\to{\textbf{Foam}}$ is an equivalence of bicategories.

Proof.

By Theorem B, $\mathcal{E}$ is a local equivalence. Hence, it is enough to show that it is essentially surjective. For that choose an object $\Sigma$ from Foam and let $k=\lfloor w(\Sigma)/2\rfloor$ , where $w(\Sigma)$ is the weight of $\Sigma$ . Then $\Sigma^{0}:=\mathcal{E}(\Sigma_{b},k)$ has the same weight. Considering $\mathbb{R}\times[0,1]$ as a disk with two boundary points removed, we can apply Lemma 3.3 to the collection of points $-\Sigma^{0}\cup\Sigma$ to obtain a web $E_{\Sigma}\colon\Sigma^{0}\to\Sigma$ with vertical lines as the underlying tangle. Another application of Lemma 3.3 combined with Proposition 2.9 shows that it is an equivalence, with its mirror image the inverse 1-morphism. ∎

We write $\mathcal{E}^{\vee}$ for the 2-functor inverse to $\mathcal{E}$ . It can be constructed explicitly like the functors $\mathcal{E}^{\vee}_{\Sigma}$ , except that the computation of signs requires not only a choice of isomorphisms between webs, but also a choice of equivalences between collections of points. For the latter one can use the following webs

which are equivalences by Proposition 2.9. They can be used to construct an explicit equivalence from a collection $\Sigma$ to $\Sigma^{0}=\mathcal{E}(\Sigma_{b},\lfloor w(\Sigma)/2\rfloor)$ by examining the points of $\Sigma$ from left to right. The details are left to the reader.

4.4 Comparison of Khovanov brackets

We finish this section by comparing two invariant complexes for a tangle $T$ : the Khovanov bracket $\llbracket T\rrbracket$ from [DrorCob], which is a formal complex of objects from ${\textbf{BN}}(\partial T)$ , and the Blanchet–Khovanov bracket $\llbracket T\rrbracket_{\mathrm{F}}$ constructed using wbes and foams instead. In what follows we recall the construction of the latter—forgetting red edges in webs and red facets in foams recovers the former.

Let $c$ be the number of crossings in $T$ , out of which $c_{+}$ are positive and $c_{-}$ are negative. The first step to construct $\llbracket T\rrbracket_{\mathrm{F}}$ is to compute the cube of resolutions of $\mathcal{I}_{\mathrm{F}}(T)$ : a commutative diagram with resolutions of $T$ at vertices of the $c$ -dimensional cube $[0,1]^{c}$ .

Namely, a vertex $\xi=(\xi_{1},\dots,\xi_{c})\in\{0,1\}^{c}$ is decorated with the web $T_{\xi}$ obtained from $T$ by replacing each $i$ -th crossing of the tangle with its resolution of type $\xi_{i}$ , as shown in Figure 5. Let $\xi^{\prime}$ be another vertex, obtained from $\xi$ by changing one coordinate from [math] to $1$ . The directed edge $\zeta\colon\xi\to\xi^{\prime}$ is decorated with the minimal foam $T_{\zeta}\colon T_{\xi}\to T_{\xi^{\prime}}$ , which is a collection of vertical facets except over the region where the two resolutions do not match; here $T_{\zeta}$ is a zip or an unzip as shown in Figure 5. It is evident that $\mathcal{I}_{\mathrm{F}}(T)$ commutes: directed paths between same vertices represent isotopic foams.

Pick a sign assignment $\epsilon$ , that is a collection of signs $\epsilon(\zeta)=\pm 1$ , one sign per edge in the cube, such that the product of signs around any square in the cube is equal to $-1$ . The standard choice is $\epsilon(\zeta)=(-1)^{s(\xi,\xi^{\prime})}$ , where $s(\xi,\xi^{\prime})$ counts $1$ ’s left to the place at which $\xi$ and $\xi^{\prime}$ disagree. Scaling each edge $\zeta$ by $\epsilon(\zeta)$ makes the cube anticommute and it can be shown that the isomorphism type of the cube is independent of the sign assignment (compare with [OddKh, Lemma 2.2] or [ChCob, Lemma 5.7]). The formal complex $\llbracket T\rrbracket_{\mathrm{F}}$ is obtained by flattening the cube along diagonals and shifting degrees accordingly. Explicitly,

[TABLE]

where $|\xi|:=\xi_{1}+\dots+\xi_{c}$ , with the differential

[TABLE]

The Khovanov bracket $\llbracket T\rrbracket$ is constructed following the same steps, except that webs and foams are replaced with flat tangles and cobordisms. In particular, one has to erase in Figure 5 the red edges in resolutions and red facets in foams.

Theorem 4.6.

The homotopy type of $\llbracket T\rrbracket_{\mathrm{F}}$ is an invariant of the tangle $T$ , strictly functorial with respect to tangle cobordism. Its image under $\mathcal{E}^{\vee}_{\partial T}$ is isomorphic to $\llbracket T\rrbracket$ .

Proof.

Following [DrorCob] one can show that $\llbracket T\rrbracket$ is functorial up to a sign and strict functoriality is shown in [Blanchet] in the case of links, i.e. when $T$ has no endpoints. From these two facts strict functoriality follows, because every tangle can be closed to a link.

To compare $\llbracket T\rrbracket_{\mathrm{F}}$ with $\llbracket T\rrbracket$ consider the cube of resolutions $\mathcal{I}_{\mathrm{F}}(T)$ constructed in ${\textbf{Foam}}(\partial T)$ and let $\mathcal{I}(T)^{\prime}$ be its image in ${\textbf{BN}}(\partial T)$ under the equivalence of categories $\mathcal{E}^{\vee}_{\partial T}$ . It differs from $\mathcal{I}(T)$ , the cube of resolutions in ${\textbf{BN}}(\partial T)$ that computes $\llbracket T\rrbracket$ , only in signs at edges. Hence, the two cubes are isomorphic and the thesis follows. ∎

*Remark 4.7**.*

The construction of $\llbracket T\rrbracket_{\mathrm{F}}$ can be easily extended to an invariant of knotted webs [Hoel] and it is conjectured to be strictly functorial with respect to foams embedded in a four dimensional space.

5 A diagrammatic TQFT on ${\textbf{Foam}}(\emptyset)$ .

The assignment of the module $\mathit{Foam}(\omega)$ to a closed web $\omega$ extends to a functor

[TABLE]

In what follows we provide a diagrammatic description of this functor by representing red-over-blue cup foams from $\mathit{Foam}(\omega)$ using certain planar diagrams and examine how the diagrams changes under action of the linear maps associated with foams.

5.1 A planar representation of cup foams

Let $\omega$ be a bounded planar web and $\omega^{+}$ its completion, which is a shading $(\omega^{+}_{r},\omega^{+}_{b})$ satisfying $\Gamma(\omega^{+}_{r},\omega^{+}_{b})=\omega$ . It is assumed that the basepoint $\ast$ marks the unbounded region, so that the region is painted white. To simplify the picture and make the web $\omega$ better visible, we do not color regions and we draw red edges as double or dashed lines depending on whether they survive or disappear after $\Gamma$ is applied, see Figure 6.

Furthermore, we allow to mark blue loops of $\omega^{+}$ with (any number of) dots. We assign to such a planar diagram a completed foam $\mathrm{cup}(\omega^{+})=\mathrm{cup}_{r}(\omega^{+})\cup\mathrm{cup}_{b}(\omega^{+})$ bounded by $\omega^{+}\!$ that satisfies the following conditions:

(CF1)

$\mathrm{cup}_{r}(\omega^{+})\subset\mathbb{R}^{2}\times[-1,0]$ and consists of disks that project injectively onto $\mathbb{R}^{2}\times\{0\}$ , 2. (CF2)

$\mathrm{cup}_{b}(\omega^{+})$ is a collection of disks such that $\mathrm{cup}_{b}(\omega^{+})\cap(\mathbb{R}^{2}\times[-1,0])=\omega^{+}_{b}\times[-1,0]$ , 3. (CF3)

each blue disk is decorated with as many dots as its boundary loop in $\omega^{+}\!$ , all placed at heights smaller than $-1$ (hence, below all red facets).

Painting the unbounded region white extends to a unique shading supported by $\mathrm{cup}(\omega^{+})$ . The resulting foam $\Gamma(\mathrm{cup}(\omega^{+}))\in\mathit{Foam}(\omega)$ is a red-over-blue cup foam. We call it the cup foam associated to $\omega^{+}$ . The following observation is an immediate consequence of Theorem 2.10.

Lemma 5.1.

Choose a completion $\omega^{+}$ of $\omega$ and consider the family of all dotted completed webs obtained from $\omega^{+}$ by placing at most one dot on each blue loop. Then the corresponding cup foams form a linear basis of $\mathit{Foam}(\omega)$ .

Notice that dots in this pictures only mark loops. In particular, moving a dot along a loop—even passing through a crossing with a red strands—does not affect the cup foam represented by the diagram.

Example 5.2.

Let $\omega$ be a blue circle. Then $\mathit{Foam}(\omega)$ is generated by two blue cups: one with and the other without a dot. These are the cup foams associated to $\omega^{+}$ when $\omega$ is oriented clockwise, because this orientation is oriented from the unbounded region, hence $\omega^{+}=\omega$ . Otherwise, $\omega^{+}$ is $\omega$ surrounded by a dashed red circle, which results in the change of the sign of the cup with a dot, see Table 1. This is consistent with the computation from Example 4.4.

5.2 Action of foams

We now provide a description of the linear maps associated to foams in terms of the dotted completed webs. In fact, it is enough to analyze the following elementary completed foams:

[TABLE]

because every foam can be decomposed into these.

Pockets and bicolored isotopies

A bicolored isotopy is a sequence of several bigon moves (3.1), which are realized by the pocket foams. When applied to a (completed) cup foam, it results in a collection of disks which may or may not be minimal, see Figure 7.

The resulting cup foam is minimal whenever the projection of a red disk is pushed through a blue arc (this creates no double points in the projection), so that the associated map takes a dotted web to the result of applying the bigon move:

[TABLE]

The shaded regions are the projections of the red disks in the corresponding completed cup foams. However, pulling the projection of a red disk off a blue arc creates double points, like in the right column of Figure 7. Indeed, the new red disk intersects the blue surface in a circle, so that either of (3.2) has to be applied. This may cost a sign, depending on the orientation of the edges:

[TABLE]

Indeed, the left moves in (5.3) and (5.4) are realized by detaching red cylinders with (2.9) whereas the right ones—by eliminating red caps with (2.16). Likewise, the relations (2.17) and (2.10) give the signs for the left and right sides respectively of both (5.5) and (5.5).

Placing a dot

Placing a dot on $\mathrm{cup}(\omega^{+})$ near the boundary violates (CF3). To obtains a minimal cup foam, the dot has to be moved down.

Let $p$ be the projection of the dot onto the horizontal plane and assume that it does not lie on a red loop. We define the nestedness $n(p)$ as the number of red loops encircling $p$ . It counts red facets below $p$ in the cup foam, hence, the number of times the dot-moving relation (2.8) has to be applied to move the dot from top to the bottom of a blue disk. Therefore, placing a dot on a blue loop results in the following map:

[TABLE]

Blue cups, caps, and saddles

Suppose now that $W$ is a completed foam with $\omega^{+}$ at its bottom and a unique critical point that lies on the blue surface. In this case $W\cup\mathrm{cup}(\omega^{+})$ is no longer a cup foam associated with the output of $W$ : to have one, the critical point of $W$ has to be slid downwards, below all red facets, and this may cost a sign. Moreover, a cap creates a sphere that has to be evaluated, whereas a split creates a neck that has to be cut.

Let $p$ be the projection of the critical point onto the horizontal plane and assume that $p\notin\omega^{+}$ . We say that a red loop $\gamma$ encircling $p$ is evenly distanced if any generic path connecting $p$ to a point $q$ from a solid (resp. dashed) red arc of $\gamma$ intersects blue circles in an even (resp. odd) number of points. Otherwise, $\gamma$ is oddly distanced. Let $s(p)$ count oddly distanced anti-clockwise and evenly distanced clockwise red loops surrounding $p$ . This corresponds to two types of red facets below $p$ : downwards oriented ones that survive in the cup foam $\Gamma(\omega^{+})$ and upwards oriented ones that are removed. These are exactly those situations, in which there is a sign in relations (2.9) and (2.10). Hence $s(p)$ determines the result of isotoping the blue critical point below all red facets. Therefore, the maps induced by critical blue points are the usual ones scaled by $(-1)^{s(p)}$ :

[TABLE]

Red cups, caps, and saddles

Placing a red cup at the top of $\mathrm{cup}(\omega^{+})$ results in a cup foam. Hence, no sign appears. Conversely, capping off an isolated red circle creates a red sphere, which can be removed by (2.6) at a cost of sign. Hence, we obtain the following maps:

[TABLE]

The behavior of merges and splits depends on whether the two red circles (those being merged or the result of a split) are nested or not. In the latter case, a merge takes a minimal cup foam to a minimal cup foam, whereas splitting a red circle creates a neck that has to be cut with (2.7) if it survives in the foam. Therefore, the correspondings maps are

[TABLE]

Merging nested red loops results in a red disk, which does not project injectively onto the horizontal plane: to fix this, the disk has to be isotoped into a croissant:

This isotopy can be described as a finger move: place your finger vertically near the saddle and move it inwards, pushing the red disk. The disk is then isotoped through any blue facet attached to a blue arc that cuts the inner red circle or a cylinder attached to a blue circle surrounded by the inner red circle:

[TABLE]

Depending on which red facets survive, each move represents two relations between foams. We leave it to the reader to check that the foams involved in the right move are always equal, whereas the left move costs a sign only in the following two configurations:

where the position of a saddle is marked with a cross. Let $c$ be the number of such configurations. Keeping in mind that a neck has to be cut in case of a split, we end up with the following formulas:

[TABLE]

Other common foams

There are other moves of interest, such as blue saddles with vertical red facets (in particular, zips and unzips) or red cups and caps that intersect vertical blue facets. All can be represented as compositions of those described above. For instance, a zip is isotopic to a pocket move followed by a saddle:

[TABLE]

As before, the shaded regions represent a projection of a red disk, and it is clear that the first move takes a basic cup foam to a basic cup foam, so that signs are governed by the second move. Therefore, the map induced by a zip is one of the following two

[TABLE]

where $s^{+}(p)$ is computed like $s(p)$ , except that we take into account the loop passing through the created red edge if it is oriented anticlockwise.

In the unzip the saddle precedes the pocket and to ensure that the latter does not affect the sign, we perform the saddle to the side of the red disk attached to the red edge:

[TABLE]

Therefore, the induced map is one of the following two:

[TABLE]

where again $s^{-}(p)$ is computed like $s(p)$ without counting the loop passing through the removed red edge.

6 The Blanchet–Khovanov invariant of tangles with balanced boundaries

Let ${\textbf{Foam}}^{\circ}$ be the subbicategory of Foam generated by balanced sequences. In what follows we construct a TQFT functor $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}\colon{\textbf{Foam}}^{\circ}\to{\textbf{Bimod}}$ . If $T$ is an oriented tangle with balanced input and output collections of points, then its resolutions are in ${\textbf{Foam}}^{\circ}\!$ , so that applying $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}$ to $\llbracket T\rrbracket_{\mathrm{F}}$ results in a chain complex of bimodules. We then show that this chain complex is isomorphic to the Khovanov’s tangle invariant [KhArcAlgebras], but it admits a strictly functorial action of tangle cobordisms.

6.1 A linear basis of webs

A web $\omega\subset\mathbb{R}\times(-\infty,0]$ is called a cup web if its underlying tangle is a cup diagram, i.e. a collection of disjoint arcs. All cup webs with the same boundary any extending the same cup diagram coincide in $\mathit{Web}$ due to Proposition 2.4 (and are isomorphic as objects of Foam). Moreover, choosing a cup web for each cup diagram results in a basis of the space of webs with given boundary, which we call a cup basis.

We describe now a particular nice cup basis of webs bounded by a balanced $\Sigma$ (see also Figure 8). Let $n$ be half of the number of blue points in $\Sigma$ (being balanced, $\Sigma$ has an even number of blue points). Proposition 3.4 provides an invertible web $E_{\Sigma}\colon\Sigma_{b}\to\Sigma$ with $2n$ vertical lines as blue edges. To obtain a cup basis, attach cup diagrams to the bottom of $E_{\Sigma}$ . In other words, the basis is the image of cup diagrams under the equivalence $\mathcal{E}_{\Sigma}$ from Section 4. We call it the red-over-blue basis of type $E_{\Sigma}$ , because all red edges in the webs appear above minima of blue cups.

6.2 Blanchet–Khovanov algebras

Fix a balanced collection of points $\Sigma$ and let $\mathcal{B}$ be a cup basis of $\mathit{Web}(\Sigma)$ .

Definition 6.1.

The Blanchet–Khovanov algebra $\mathfrak{W}^{\mathcal{B}}$ associated with $\mathcal{B}$ is the direct sum of spaces of foams with corners

[TABLE]

with multiplication given by the composition (and zero if foams cannot be composed).

*Remark 6.2**.*

The above algebra appeared first in [WebAlgebras] for $\Sigma$ a collection of positively oriented blue points followed by negatively oriented red points, the latter drawn in [WebAlgebras] at the bottom.

Choose a completion $a^{+}$ for any cup web $a$ and write $a^{!}$ (resp. $(a^{+})^{!}$ ) for the result of reflecting $a$ (resp. $a^{+}$ ) along the horizontal line and reversing orientation of edges. Using the natural isomorphisms $\mathrm{Hom}_{{\textbf{Foam}}(\Sigma)}(a,b)\cong\mathit{Foam}(b^{!}a)\{n\}$ we can represent elements of the algebra by dotted completed webs, in which case the multiplication is induced from the family of generalized saddles

[TABLE]

each consisting of the identity foams $(c^{+})^{!}\times[0,1]$ and $a^{+}\times[0,1]$ glued to the half-rotation of $b^{+}$ around the boundary line. These foams take a particularly nice form when $\mathcal{B}$ is a red-over-blue basis, as they involve then only three types of moves:

•

merging (5.10) and splitting (5.11) blue loops at points outside of all red circles,

•

merging unnested red loops (5.14), and

•

removing bigons external to the projection of red disks (5.1).

Hence, the product of two dotted diagrams is a positive linear combination of other diagrams.

Corollary 6.3.

The algebra $\mathfrak{W}^{\mathcal{B}}$ admits a positive basis.

When $\Sigma$ is a collection of blue points oriented in the alternating way and $\mathcal{B}$ consists of oriented cup diagrams (i.e. webs with no red edges), then $\mathfrak{W}^{\mathcal{B}}$ coincides with the arc algebra $H^{n}$ from [KhArcAlgebras]. Indeed, $\mathfrak{W}^{\mathcal{B}}$ is the image of $H^{n}$ under the embedding of bicategories $\mathcal{E}\colon{\textbf{BN}}\to{\textbf{Foam}}$ . However, when $\mathcal{B}$ is not a red-over-blue basis, then the generalized saddles (6.2) may involve moves on red arcs that cost a sign, such as splits (5.15) or nested saddles (5.16) and (5.17). Hence, cup foams do not constitute a positive basis of the algebra in such case. Yet, it is still isomorphic to the arc algebra.

Theorem 6.4.

Let $\Sigma$ be a balanced collection of points with $2n$ blue points. Then there is an algebra isomorphism $\mathfrak{W}^{\mathcal{B}}\cong H^{n}$ for any cup basis $\mathcal{B}$ of webs bounded by $\Sigma$ . When $\mathcal{B}$ is a red-over-blue basis, then the isomorphism simply forgets red facets of basic cup foams.

Proof.

Assume first that $\mathcal{B}$ is a red-over-blue basis of type $E_{\Sigma}$ . Then $\mathfrak{W}^{\mathcal{B}}$ is the image of $H^{n}$ under the equivalence of categories $\mathcal{E}_{\Sigma}$ , which equips a collection of dotted cups with its standard orientation. The inverse of $E_{\Sigma}$ simply forgets red edges in webs and red facets in foams. Hence, the thesis follows.

Let now $\mathcal{B}^{\prime}$ be any cup basis and pick for each cup web $a^{\prime}\in\mathcal{B}^{\prime}$ the isomorphic cup web $a\in\mathcal{B}$ , an invertible foam $I_{a}\in{\textbf{Foam}}(a,a^{\prime})$ , and $s(a)=\pm 1$ such that $\mathrlap{\phantom{I}{}^{!}}I_{a}\,I_{a}=s(a)\,\omega_{a}\times[0,1]$ . Then the collection of linear isomorphisms

[TABLE]

constitute an isomorphism of algebras $\mathfrak{W}^{\mathcal{B}^{\prime}}\cong\mathfrak{W}^{\mathcal{B}}\!$ , where the latter is isomorphic to $H^{n}\!$ . ∎

Example 6.5.

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}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.5pt}{0.0pt}\pgfsys@curveto{2.5pt}{1.38072pt}{1.38072pt}{2.5pt}{0.0pt}{2.5pt}\pgfsys@curveto{-1.38072pt}{2.5pt}{-2.5pt}{1.38072pt}{-2.5pt}{0.0pt}\pgfsys@curveto{-2.5pt}{-1.38072pt}{-1.38072pt}{-2.5pt}{0.0pt}{-2.5pt}\pgfsys@curveto{1.38072pt}{-2.5pt}{2.5pt}{-1.38072pt}{2.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.99997pt}{4.29134pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8,0,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0.2}\pgfsys@color@rgb@stroke{0.8}{0}{0.2}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8}{0}{0.2}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8,0,0.2}\color[rgb]{0.8,0,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0.2}$ \scriptstyle-\vphantom{+} $}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ . Then the cup basis $\mathcal{B}$ consists of two elements

$\scriptstyle+$$\scriptstyle+$$\scriptstyle-\vphantom{+}$$\scriptstyle+$$\scriptstyle-\vphantom{+}$

and $\scriptstyle+$$\scriptstyle+$$\scriptstyle-\vphantom{+}$$\scriptstyle+$$\scriptstyle-\vphantom{+}$

that form four pairs: two of them have two blue loops and each of the other two has one blue loop. Hence, $\dim\mathfrak{W}^{\mathcal{B}}=12$ . The multiplication looks a lot like in $H^{2}\!$ . For instance,

[TABLE]

Erasing red edges recovers the usual diagrammatic calculus of $H^{2}\!$ .

Example 6.6.

Recall the Blanchet–Khovanov algebra from [WebAlgebras]: it is defined using webs that have only vertical red edges, $2n$ positive blue endpoints at the top and positive $n$ red endpoints at the bottom. We call them here EST webs. To fit this construction into our framework, we move the bottom endpoints rightwards and to the top by appending a collection of nested red cups (see Figure 9).

Contrary to the case of red-over-blue bases, minima of red cups in EST webs appear below all blue cups, because of which the formula for multiplication involves lots of signs. Yet Theorem 6.4 provides a direct isomorphism between this algebra and the arc algebra. Such an isomorphism was explicitly constructed in [WebVsArcAlgebra] by providing a sign for each generator of the algebra, then checking directly that these signs result in a homomorphism of algebras.

6.3 Blanchet–Khovanov bimodules

Pick now two balanced collections $\Sigma$ and $\Sigma^{\prime}$ with cup bases $\mathcal{B}$ and $\mathcal{B}^{\prime}$ respectively. We assign to a web $\omega\colon\Sigma\to\Sigma^{\prime}$ its Blanchet–Khovanov bimodule $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(\omega)$ , which is the $(\mathfrak{W}^{\mathcal{B}},\mathfrak{W}^{\mathcal{B}^{\prime}})$ -bimodule

[TABLE]

where ${\textbf{Foam}}(a,\omega,b)$ is the space of foams bounded by $b^{!}\cup\omega\cup a$ and seen as foams in a cube with $\omega$ at the top facet, whereas $a$ and $b$ lie on opposite vertical facets (see Figure 10).

The algebras $\mathfrak{W}^{\mathcal{B}}$ and $\mathfrak{W}^{\mathcal{B}^{\prime}}$ act on the left and on the right respectively, and there is a diagrammatic presentation of this bimodule as explained in Section 5. Moreover, placing a foam $S\in{\textbf{Foam}}(\omega,\omega^{\prime})$ on top results in a bimodule map $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(S)\colon\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(\omega)\to\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(\omega^{\prime})$ . We check that $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(S)=\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(S^{\prime})$ if the two foams coincide in Foam, so that the map is well-defined. In particular, $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(\omega)\cong\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(\omega^{\prime})$ if $S$ is invertible. Finally, horizontal composition of foams induces a canonical homomorphism of graded bimodules

[TABLE]

for any pair of composable webs $\omega\colon\Sigma\to\Sigma^{\prime}$ and $\omega^{\prime}\colon\Sigma^{\prime}\to\Sigma^{\prime\prime}$ . The proof of [KhArcAlgebras, Theorem 1] can be adapted to our framework to show that (6.4) is an isomorphism.

*Remark 6.7**.*

Contrary to the case of Blanchet–Khovanov algebras, the isomorphism (6.4) may not take a pair of cup foams into a positive combination of cup foams. When using the diagrammatics of completed webs, (6.4) is induced by a collection of generalized saddles, the description of which—contrary to the case of algebras—may involve moves on red loops that cost a sign, such as (5.15)–(5.17). However, this is not the case when both webs have only blue endpoints, oriented in an alternating way—in this case all red loops lie inside the webs and are not affected when webs are composed.

As in the case of algebras, $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(\omega)$ coincides with the arc bimodule $\mathcal{F}_{\!\textit{Kh}}^{\circ}(\omega)$ defined in [KhArcAlgebras] when $\omega$ is a standardly oriented flat tangle and both $\mathcal{B}$ and $\mathcal{B}^{\prime}$ are collections of cup diagrams. Although in general $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(\omega)$ is not a priori a bimodule over arc algebras, it can be made such through the algebra isomorphisms $H^{n}\cong\mathfrak{W}^{\mathcal{B}}$ and $H^{n^{\prime}}\cong\mathfrak{W}^{\mathcal{B}^{\prime}}$ provided by Theorem 6.4. Hence, it makes sense to compare $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(\omega)$ with $\mathcal{F}_{\!\textit{Kh}}^{\circ}(\omega_{b})$ .

Theorem 6.8.

Let $\omega\colon\Sigma\to\Sigma^{\prime}$ be a web between balanced collections of points with $2n$ and $2n^{\prime}$ blue points respectively. Then there is an isomorphism of $(H^{n},H^{n^{\prime}})$ –bimodules $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(\omega)\cong\mathcal{F}_{\!\textit{Kh}}^{\circ}(\omega_{b})$ . The isomorphism simply forgets red facets of basic cup foams when $\mathcal{B}$ and $\mathcal{B}^{\prime}$ are red-over-blue web bases.

Proof.

Assume first that $\mathcal{B}$ and $\mathcal{B}^{\prime}$ are red-over-blue cup bases of types $E_{\Sigma}$ and $E_{\Sigma^{\prime}}$ respectively. Fix a foam $I_{\omega}$ in a cube with vertical rectangles as blue facets, bounded by $\omega_{b}$ and $\omega$ at the bottom and top facets, and with $E_{\Sigma}$ and $E_{\Sigma^{\prime}}$ at appropriate vertical facets. Placing it on top of an element from ${\textbf{Foam}}(a,\omega_{b},b)$ results in a $\Bbbk$ –linear isomorphism ${\textbf{Foam}}(a,\omega_{b},b)\cong{\textbf{Foam}}(E_{\Sigma}\cup a,\omega,E_{\Sigma^{\prime}}\cup b)$ . It is straightforward to check that these isomorphisms are compatible with the action of the arc algebras, so that they constitute an isomorphism of bimodules $\mathcal{F}_{\!\textit{Kh}}^{\circ}(\omega_{b})\cong\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(\omega)$ ; it takes a collection of dotted cups to a basic cup foam. Forgetting red facets is the inverse map.

The general case is reduced to the above as in the proof of Theorem 6.4: choose a collection of invertible foams, one per $a\in\mathcal{B}$ and one per $b\in\mathcal{B}^{\prime}$ , and glue them to the sides of foams generating $\mathcal{F}_{\!\textit{Kh}}^{\circ}(\omega)$ . ∎

*Remark 6.9**.*

When $\mathcal{B}$ is a red-over-blue basis of type $E_{\Sigma}$ , then the action of $H^{n}$ can be understood pictorially as follows: a dotted surface $S\in H^{n}$ is standardly oriented and combined with $E_{\Sigma}\times[0,1]$ before acting on $\mathcal{F}_{\!\textit{Kh}}^{\circ}(\omega)$ . The same applies to $H^{n^{\prime}}$ if $\mathcal{B}^{\prime}$ is a red-over-blue basis.

We say that a linear basis $\{x_{1},\dots,x_{d}\}$ of an $(A,B)$ -bimodule is positive with respect to bases $\{a_{i}\}$ of $A$ and $\{b_{j}\}$ of $B$ , when each $a_{i}x_{k}$ and $x_{k}b_{j}$ has positive coefficients in this basis. Because dotted cups constitute a positive basis of arc bimodules, Theorem 6.8 implies the existence of a positive basis for Blanchet–Khovanov bimodules.

Corollary 6.10.

Suppose that both $\mathcal{B}$ and $\mathcal{B}^{\prime}\!$ are red-over-blue cup bases of webs. Then basic cup foams constitute a positive basis for $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(\omega)$ .

Although the formulas for the actions of the algebras on a Blanchet–Khovanov bimodule involve no signs when red-over-blue bases as used, this is not the case for action of foams: the following squares commutes only up to a sign

[TABLE]

where we abuse the notation and denote the isomorphism from the proof of Theorem 6.8 by the same symbol $I_{\omega}$ as the foam used to construct it. However, the sign does not depend on the direct summand of the bimodule: it is determined by the configuration of red loops (see Section 5) and the configuration is the same for all closures $b^{!}\omega a$ .

6.4 A functorial homology for tangles with balanced boundaries

The previous sections describe a morphism fo bicategories $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}\colon{\textbf{Foam}}^{\circ}\to{\textbf{Bimod}}$ , which we extend naturally to $\mathit{Com}_{\!\raisebox{0.90417pt}{$ \scriptstyle/ $}\mkern-2.0muh}({\textbf{Foam}}^{\circ})$ . As mentioned in the introduction, we can apply it to the formal bracket $\llbracket T\rrbracket_{\mathrm{F}}$ of a tangle $T$ with balanced input and output, producing a chain complex $\mathit{C}_{\mathfrak{W}}(T)$ . Invariance and functoriality of the bracket implies that the homotopy type of $\mathit{C}_{\mathfrak{W}}(T)$ is an invariant of the tangle $T$ that is functorial with respect to tangle cobordisms.

Theorem C.

The 2-functor $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}$ is equivalent to $\mathcal{F}_{\!\textit{Kh}}^{\circ}\circ\mathcal{E}^{\vee}\!$ . In particular, the complexes $\mathit{C}_{\mathfrak{W}}(T)$ and $\mathit{CKh}(T)$ are isomorphic for any tangle $T\kern-0.5pt$ with balanced input and output.

Proof.

The two functors coincide on objects by Theorem 6.4 and on 1-morphisms by Theorem 6.8. Furthermore, the collection of isomorphisms $I_{\omega}$ is natural in $\omega$ , because the square (6.5) commutes when $S_{b}$ is replaced with $\mathcal{E}^{\vee}(S)$ . Indeed, the sign relating $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\circ}(S)\circ I_{\omega}$ with $I_{\omega^{\prime}}\circ\mathcal{F}_{\!\textit{Kh}}^{\circ}(S_{b})$ is exactly the one provided by $\mathcal{E}^{\vee}$ . The last statement is a direct consequence of Theorem 4.6. ∎

7 Subquotient algebras and an invariant for all tangles

Inspired by [ChenKhov] we use the TQFT from the previous section to define a family of 2-functors $\mathcal{F}_{\mkern-3.0mu\mathit{w}}^{\lambda}\colon{\textbf{Foam}}\to{\textbf{Bimod}}$ parametrized by $\lambda\in\mathbb{Z}$ , which are defined on the whole bicategory of foams. As before, these 2-functors lead to invariant chain complexes for tangles that are strictly functorial versions of the Chen–Khovanov tangle invariants. Contrary to the previous sections, we assume here that $h=t=0$ . In particular, a foam vanishes when it has a blue facet with two dots.

7.1 Balancing

Suppose that $\Sigma$ has $m$ red and $n$ blue points and choose $0\leqslant k\leqslant n$ . We say that a sequence $\Sigma^{\circ}$ on a line with platforms is a balancing of $\Sigma$ of type $n-2k$ if it is balanced and obtained from $\Sigma$ by placing $\ell$ and $r$ blue points to the left and right of $\Sigma$ respectively, where $r-\ell=n-2k$ . We say that the extra points lie on platforms, which are drawn as dashed lines. In what follows we describe two methods to balance a given sequence, see Figure 11.

The mirror balancing $\Sigma^{k,n-k}_{mir}$ of $\Sigma$ of type $\lambda=n-2k$ is constructed as follows. First, replace each red point by two blue points oriented the same way and call the new sequence $\Sigma^{\prime}$ . Then $\Sigma^{k,n-k}_{mir}$ is obtained from $\Sigma$ by placing the first $k+m$ points of $\Sigma^{\prime}$ on the left and the remaining ones on the right platform, both in the reversed order; we also reflect the orientation of points (compare this with Figure 11). It is a balanced sequence, which is an alternating sequence of blue points if $\Sigma$ is such. However, it depends heavily on the orientations of points of $\Sigma$ . The next construction does not share his drawback.

The canonical balancing $\Sigma^{k,n-k}_{can}$ of type $\lambda=n-2k$ is constructed again by placing $k+m$ points on the left and $n-k+m$ points on the right platform, except that now we order the points in a way, such that, when read from left to right, positive points on each platform appear first. Moreover, we want the minimal number of negative (resp. positive) points on the left (resp. right) platform. This leads to one of the following distributions, depending on the total weight $w=w(\Sigma)$ of the sequence $\Sigma$ .

Case $|w|\geqslant|\lambda|$

[TABLE]

Case $|w|<\lambda$

[TABLE]

Case $|w|<-\lambda$

[TABLE]

We check directly that in each case we obtain a balanced sequence with at most $k+m$ negative and at most $n-k+m$ positive points on the left and right platform respectively.

*Remark 7.1**.*

The distribution of points on platforms in $\Sigma^{k,n-k}_{can}$ depends only on the total weight $w$ of the sequence and the type $\lambda$ of the balancing, but not directly on the number of points nor their orientation. This is why we call it canonical.

7.2 Webs and foams with platforms

We now allow foams to meet the side vertical facets of the ambient cube in collections of horizontal blue lines. More precisely, fix a web $\omega\colon\Sigma_{0}\to\Sigma_{1}$ together with balanced collections $\Sigma_{0}^{\circ}$ and $\Sigma_{1}^{\circ}$ , such that the first $\ell$ and last $r$ points of both $\Sigma_{1}^{\circ}$ and $\Sigma_{1}^{\circ}$ are blue, oriented the same way, and removing them recovers $\Sigma_{1}$ and $\Sigma_{2}$ respectively. Given cup webs $a$ and $b$ bounded by $\Sigma_{0}^{\circ}$ and $\Sigma_{1}^{\circ}$ respectively, we write $\mskip-2.0mu{\mathrlap{\widetilde{\phantom{\textbf{Fo}}}}}\mskip 2.0mu{\textbf{Foam}}^{(\ell,r)}(a,\omega,b)$ for the space of foams embedded in a cube with the following boundary:

•

the web $\omega$ at the top facet of the cube,

•

$\ell$ and $r$ horizontal blue lines at the vertical facets to the left and to the right of $\omega$ respectively, and

•

the cup webs $a$ and $b$ at the vertical facets attached to the input and output of $\omega$ .

Figure 12 provides examples of such foams.

We say that such a foam is violating if it has a connected component that either

•

meets a platform in more than one lines, or

•

intersects a platform and carries a dot.

It is straightforward to check that the property of being a violating foam is preserved by foam relations. Hence, violating foams generate a linear subspace of $\mskip-2.0mu{\mathrlap{\widetilde{\phantom{\textbf{Fo}}}}}\mskip 2.0mu{\textbf{Foam}}^{(\ell,r)}(a,\omega,b)$ . We write ${\textbf{Foam}}^{(\ell,r)}(a,\omega,b)$ for the quotient space, or simply ${\textbf{Foam}}^{(\ell,r)}(a,b)$ when $\omega$ is the identity web.

Gluing foams horizontally results in a linear map

[TABLE]

and it is straightforward to notice that a foam $S^{\prime}S$ is violating when either $S$ or $S^{\prime}$ is a violating foam. Hence, there is an induced linear map

[TABLE]

Consider now webs with platforms as discussed in Section 7.1. Their blue arcs fall into three families visualized in Figure LABEL:fig:blue-arc-types:

•

inner arcs, with at least one endpoint not on a platform,

•

outer arcs, with each endpoint on a different platform, and,

•

violating arcs, with both endpoints on the same platform.

Webs with no violating arcs and no red endpoints on platforms are admissible. Outer arcs of an admissible web are nested one in another and the most nested one of them encloses all inner arcs. Notice that ${\textbf{Foam}}^{(\ell,r)}(a,\omega,b)=0$ when either $a$ or $b$ has a violating arc.

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On the functoriality of sl2\mathfrak{sl}_{2}sl2​ tangle homology

Abstract

Contents

1 Introduction

1.1 The equivalence of foams and Bar-Natan cobordisms

1.1.1 Main tools: shadings and bicolored isotopies

1.2 Functorial tangle homology

1.3 Functoriality of quantized annular Khovanov homology

Corollary E**.**

Corollary F**.**

1.4 Further generalizations

1.5 Organization of the paper

1.6 Conventions and notation

1.7 Acknowledgments

2 Main players

2.1 Webs

Remark 2.1*.*

Definition 2.2**.**

Exercise 2.3**.**

Proposition 2.4**.**

Lemma 2.5**.**

Proof.

2.2 Foams

Definition 2.6**.**

Remark 2.7*.*

Exercise 2.8**.**

Proposition 2.9**.**

Theorem 2.10**.**

2.3 Decategorification

Theorem 2.11**.**

Proof.

2.4 Higher structures

2.5 Blanchet evaluation formula

Example 2.12**.**

Theorem 2.13** (cp. [Blanchet]).**

Proof.

3 Shadings and a basis of foams

3.1 Shadings and trivalent manifolds

Lemma 3.1**.**

Proof.

Remark 3.2*.*

Lemma 3.3**.**

Proof.

Proposition 3.4**.**

Proof.

3.2 Bicolored isotopies

Lemma 3.5**.**

Lemma 3.6** (Bicolored Isotopy).**

Proof.

Proof of Proposition 2.4.

Proof of Proposition 2.9.

3.3 Cup foams

Lemma 3.7**.**

Proof.

Lemma 3.8**.**

Proof.

Proof of Theorem 2.10.

4 Equivalences of foam and cobordism categories

4.1 Embedding cobordisms into foams

Lemma 4.1**.**

Proof.

Remark 4.2*.*

Theorem B**.**

Proof.

4.2 A coherent way to forget red facets

Proposition 4.3**.**

Proof.

Example 4.4**.**

Remark 4.5*.*

4.3 An equivalence of bicategories

Theorem A**.**

Proof.

4.4 Comparison of Khovanov brackets

On the functoriality of $\mathfrak{sl}_{2}$ tangle homology

Corollary E.

Corollary F.

*Remark 2.1**.*

Definition 2.2.

Exercise 2.3.

Proposition 2.4.

Lemma 2.5.

Definition 2.6.

*Remark 2.7**.*

Exercise 2.8.

Proposition 2.9.

Theorem 2.10.

Theorem 2.11.

Example 2.12.

Theorem 2.13 (cp. [Blanchet]).

Lemma 3.1.

*Remark 3.2**.*

Lemma 3.3.

Proposition 3.4.

Lemma 3.5.

Lemma 3.6 (Bicolored Isotopy).

Lemma 3.7.

Lemma 3.8.

Lemma 4.1.

*Remark 4.2**.*

Theorem B.

Proposition 4.3.

Example 4.4.

*Remark 4.5**.*

Theorem A.

Theorem 4.6.

*Remark 4.7**.*

5 A diagrammatic TQFT on ${\textbf{Foam}}(\emptyset)$ .

Lemma 5.1.

Example 5.2.

Definition 6.1.

*Remark 6.2**.*

Corollary 6.3.

Theorem 6.4.

Example 6.5.

Example 6.6.

*Remark 6.7**.*

Theorem 6.8.

*Remark 6.9**.*

Corollary 6.10.

Theorem C.

*Remark 7.1**.*