Relationship between quandle shadow cocycle invariants and Vassiliev invariants

TL;DR

This paper establishes a connection between quandle shadow cocycle invariants and Vassiliev invariants, showing that the coefficients of certain quandle invariants are Vassiliev invariants for braids.

Contribution

It proves that the coefficients of the finite perturbative expansion of specific quandle shadow cocycle invariants are Vassiliev invariants, answering a question posed by T. Ohtsuki.

Findings

Coefficients of quandle shadow cocycle invariants are Vassiliev invariants.

The result applies to invariants defined by $( ext{Z}/p ext{Z})$-Laurent polynomial quandles.

Establishes a link between quantum invariants and finite-type invariants.

Abstract

As one of the problems in his list [20], T. Ohtsuki proposed to study relations between quandle cocycle invariants and quantum invariants. The aim of this paper is to answer one of those questions. We prove that the coefficient of the finite perturbative expansion of the quandle shadow cocycle invariant defined by $(\Z /p\Z)$-Laurent polynomial quandle is Vassiliev invariant for any braids.

Equations

$$x\ast y=\omega x+(1-\omega )y$$

$$C_Q^n(X;A):=\{f:X^n\to A|f(x_1,\dots ,x_n)=0\mathrm{when}{x}_i=x_{i+1}\mathrm{for}\mathrm{some}i\}.$$

$$g_{a_1,\dots ,a_n}(x_1,\dots ,x_n)=\{\begin{array}{ll}1& \text{mbox}if(x_1,\dots ,x_n)=(a_1,\cdots ,a_n),\\ 0& \text{mbox}otherwise,\end{array}$$

$$g_{a_{1},\dots,a_{n}}(x_{1},\dots,x_{n})\,=\,\prod_{i=1}^{n}\,\bigl{(}\prod_{b_{i}\in\mathop{\mathbb{F}_{q}}\nolimits:\,\,b_{i}\neq a_{i}}\frac{x_{i}-b_{i}}{a_{i}-b_{i}}\bigr{)}.$$

$$C_{Q}^{n}:={\rm span}_{\mathop{\mathbb{F}_{q}}\nolimits}\big{\{}U_{1}^{i_{1}}\cdots U_{n}^{i_{n}}\ \big{|}\ 1\leq i_{j}\leq q-1,\hskip 5.0pt0\leq i_{n}\leq q-1\big{\}}.$$

$${\delta }_n(f)(U_1,\dots ,U_{n+1})=\sum _{i=1}^n(-1{)}^{i-1}$$

$$\displaystyle\quad-f(U_{1},\dots,U_{i-1},U_{i}+U_{i+1},U_{i+2},\dots,U_{n+1})\Big{)}.$$

$$\big{\{}U_{1}^{p^{v}}U_{2}^{p^{u}}\ \big{|}\ \ \omega^{p^{v}+p^{u}}=1,\ \ 0\leq v<u<m\big{\}}.$$

$$\displaystyle I_{1}:=\big{\{}U_{1}^{p^{v}}U_{2}^{p^{u}}U_{3}^{p^{t}}\ \big{|}\ \ \omega^{p^{v}+p^{u}+p^{t}}=1,\ \ 0\leq v<u<t<m\big{\}},$$

$$\displaystyle I_{2}:=\big{\{}\big{(}\chi(\omega U_{1},U_{2})-\chi(U_{1},U_{2})\big{)}^{p^{u}}\cdot U_{3}^{p^{t}}\ \big{|}\ \ \omega^{p^{u+1}+p^{t}}=1,\ \ 0\leq u<t<m\big{\}},$$

$$\displaystyle I_{3}:=\big{\{}U_{1}^{p^{v}}\big{(}\chi(U_{2},U_{3})-\chi(U_{2},\omega^{-1}U_{3})\big{)}^{p^{t}}\ \big{|}\ \ \omega^{p^{v}+p^{t+1}}=1,\ \ 0\leq v\leq t<m\big{\}},$$

$$\displaystyle I_{0}:=\big{\{}U_{1}^{p^{v}}U_{2}^{p^{u}}\ \big{|}\ \ \omega^{p^{v}+p^{u}}=1,\ \ 0\leq v<u<m\big{\}},$$

$$\chi(U_{i},U_{i+1}):=\sum_{j=1}^{p-1}(-1)^{j-1}j^{-1}U_{i}^{p-j}U_{i+1}^{j}\equiv\frac{1}{p}\Bigl{(}(U_{i}+U_{i+1})^{p}-U_{i}^{p}-U_{i+1}^{p}\Bigr{)}\pmod{p}.$$

$$I_4:=I_{\text{mbox}4-1}\cup I_{\text{mbox}4-2}\cup I_{\text{mbox}4-3}\cup I_{\text{mbox}4-4}\cup I_{\text{mbox}4-5},$$

$$\displaystyle I_{\mbox{\scriptsize 4-1}}=\big{\{}U_{1}^{p^{v}}U_{2}^{p^{u}+p^{t}}U_{3}^{p^{s}}\ \big{|}\ \omega^{p^{v}+p^{u}}=1,\ \ (\ref{42})\big{\}},$$

$$\displaystyle I_{\mbox{\scriptsize 4-2}}=\Bigg{\{}\begin{array}[]{l}U_{1}^{p^{v}}U_{2}^{p^{u}+p^{t}}U_{3}^{p^{s}}-U_{1}^{p^{u}}U_{2}^{p^{v}+p^{s}}U_{3}^{p^{t}}\\ -\frac{1-\omega^{p^{v}+p^{u}}}{\omega^{p^{u}}-1}\big{(}U_{1}^{p^{v}}U_{2}^{p^{u}}U_{3}^{p^{t}+p^{s}}-U_{1}^{p^{v}+p^{u}}U_{2}^{p^{s}}U_{3}^{p^{t}}\big{)}\end{array}\ \Bigg{|}\ \omega^{p^{v}+p^{u}}\neq 1,\ \ t>s,\ \ (\ref{42})\Bigg{\}},$$

$$\displaystyle I_{\mbox{\scriptsize 4-3}}=\big{\{}U_{1}^{p^{v}}U_{2}^{p^{u}+p^{t}}U_{3}^{p^{s}}-2^{-1}(1-\omega^{-p^{t}})U_{1}^{p^{v}}U_{2}^{p^{u}}U_{3}^{p^{t}+p^{s}}\ \big{|}\ p\neq 2,\ \ \omega^{p^{v}+p^{u}}\neq 1,\ \ t=s,\ \ (\ref{42})\big{\}},$$

$$\displaystyle I_{\mbox{\scriptsize 4-4}}=\Bigg{\{}\begin{array}[]{l}U_{1}^{p^{v}}U_{2}^{p^{u}+p^{t}}U_{3}^{p^{s}}+U_{1}^{p^{u}}U_{2}^{p^{v}+p^{t}}U_{3}^{p^{s}}\\ \quad-\frac{1-\omega^{2p^{v}}}{\omega^{p^{v}}-1}U_{1}^{p^{v}+p^{u}}U_{2}^{p^{t}}U_{3}^{p^{s}}\end{array}\ \Bigg{|}\ \begin{array}[]{l}p\neq 2,\ \ \omega^{p^{v}+p^{u}}\neq 1,\ \ u\leq v<t<s,\\ \omega^{p^{v}}=\omega^{p^{u}},\ \ (\ref{42})\end{array}\Bigg{\}},$$

$$\displaystyle I_{\mbox{\scriptsize 4-5}}=\Bigg{\{}\begin{array}[]{l}U_{1}^{p^{v}}U_{2}^{p^{u}+p^{t}}U_{3}^{p^{s}}+U_{1}^{p^{u}}U_{2}^{p^{v}+p^{t}}U_{3}^{p^{s}}\\ \quad-\frac{1-\omega^{2p^{v}}}{\omega^{p^{v}}-1}U_{1}^{p^{v}+p^{u}}U_{2}^{p^{t}}U_{3}^{p^{s}}\end{array}\ \Bigg{|}\ \begin{array}[]{l}p=2,\ \ \omega^{p^{v}+p^{u}}\neq 1,\ \ u<v<t<s,\\ \omega^{p^{v}}=\omega^{p^{u}},\ \ (\ref{42})\end{array}\Bigg{\}},$$

$$u\le t,v<t<m,u<s<m,{\omega }^{p^v+p^t}={\omega }^{p^u+p^s}=1.$$

$$W\Biggl{(}\parbox{49.8663pt}{\leavevmode\hbox{\includegraphics[width=50.0pt]{coloring4.eps}}}\Biggr{)}=f(x,y)\in A$$

$$W\Biggl{(}\parbox{49.8663pt}{\leavevmode\hbox{\includegraphics[width=50.0pt]{coloring5.eps}}}\Biggr{)}=f(x,y)^{-1}\in A$$

$$\sum _C\prod _x{W}_f(x;C)\in \mathbb{Z}[A],$$

$${\Phi }_f(L)=\sum _C\prod _x{W}_f(x;C)\in \mathbb{Z}[A].$$

$$W\Biggl{(}\parbox{49.8663pt}{\leavevmode\hbox{\includegraphics[width=50.0pt]{coloring6.eps}}}\Biggr{)}=\phi(z,x,y)\in A$$

$$W\Biggl{(}\parbox{49.8663pt}{\leavevmode\hbox{\includegraphics[width=50.0pt]{coloring7.eps}}}\Biggr{)}=\phi(z,x,y)^{-1}\in A$$

$$\sum _C\prod _x{W}_{\varphi }(x;C)\in \mathbb{Z}[A],$$

$${\Phi }_{\varphi }(L)=\sum _C\prod _x{W}_{\varphi }(x;C)\in \mathbb{Z}[A].$$

$${\psi }_n({\sigma }_i)=(i{d}_V{)}^{\otimes (i-1)}\otimes R\otimes (i{d}_V{)}^{\otimes (n-i-1)}.$$

$$(R\otimes i{d}_V)(i{d}_V\otimes R)(R\otimes i{d}_V)=(i{d}_V\otimes R)(R\otimes i{d}_V)(i{d}_V\otimes R).$$

$$R_{x^{\prime }{y}^{\prime }}^{xy}=\{\begin{array}{ll}f(x,y),& (\text{when x′=y and y′=x∗y}),\\ 0,& (otherwise).\end{array}$$

$$t^{kp+i}{|}_{t=e^{a\hslash }}=\sum _{d=0}^{\infty }\frac{1}{d!}(kp+i{)}^d(a\hslash {)}^d\equiv \sum _{d=0}^{\infty }\frac{1}{d!}{i}^d(a\hslash {)}^d=t^i{|}_{t=e^{a\hslash }}.$$

$${\Phi }_f(L)\in \mathbb{Z}[{\mathbb{F}}_q]\cong \mathbb{Z}[t]/(t^p-1,g(t),h_1(\omega ),\dots ,h_{\ell }(\omega )).$$

$${\Phi }_f(L){|}_{t=e^{a\hslash },\omega =e^{b\hslash }}\mapsto \sum _{d=0}^{\infty }d!u_d(L){\hslash }^d\in \mathbb{Z}/p\mathbb{Z}[[\hslash ]].$$

Full text

**Keywords

Vassiliev invariants, Quandle (shadow) cocycle invariants.**

1 Introduction

We show that there exist links such that Vassiliev invariants can be deduced from quandle cocycle invariants by using Alexander quandles. Also, many quandle (shadow) cocycle invariants by using Alexander quandles is to study the space of Vassiliev invariants; all coefficients of quandle (shadow) cocycle invariants by using Alexander quandles after a suitable change of variables are Vassiliev invariants in ${\mathbb{Z}}/p{\mathbb{Z}}$.

An $R$-matrix whose components are appropriately set for the values of a quandle cocycle satisfies the set-theoretic Yang–Baxter equation, and the values of the operator invariants calculated using the $R$-matrix equals those of the quandle cocycle invariants [15, 17]. And we can obtain Vassiliev invariants in the case where quandle (shadow) cocycles are calculated using trivial quandles [1]. The relation between (shadow) quandle cocycle invariants and operator invariants has been clarified using only the above-mentioned method. However, we successfully deduced Vassiliev invariants from quandle shadow cocycle invariants. As shown in the figure 1 below, we find a relation between the $2$ types of invariants described above, which were considered to have no relation to each other. In particular, we find that in the case of Alexander quandles, Vassiliev invariants can be obtained using quandle (shadow) cocycle invariants. This is expected to have applications to surface links and low-dimensional manifolds in the future.

In Section 2, we review the quandle (shadow) cocycle invariants and Vassiliev invariants. A quandle is a set with a binary operation satisfying certain axioms, which is analogous to a group with conjugation. The cohomology groups of quandles were introduced by Carter, Jelsovsky, Kamada, Langford, and Saito [3] as an analogy of group cohomology. We denote the $n$-th cohomology group of quandle $X$ with coefficient group $A$ by $H^{n}_{Q}(X;A)$. Furthermore, the (shadow) quandle cocycle invariants of classical links were defined using $2$-cocycles or $3$-cocycles of the cohomology groups of quandles [3, 4].

In Section 3, we prove that Vassiliev invariants can be deduced using the quandle cocycle invariants of Alexander quandles.

2 Quandle cocycle invariants and Vassiliev invariants

A quandle is a set, $X$, with a binary operation, $\ast:X\times X\to X$, such that the following three conditions are satisfied:

1. (i)

For any $a\in X$, $a\ast a=a.$ 2. (ii)

For any $a,b\in X$, there exists a unique $c\in X$ such that $a=c\ast b$. 3. (iii)

For any $a,b,c\in X$, $(a\ast b)\ast c=(a\ast c)\ast(b\ast c).$

Let ${\mathbb{Z}}[\omega^{\pm}]$ be the Laurent polynomial ring and $J$ be an ideal of ${\mathbb{Z}}[\omega^{\pm}]$. We fix an invertible element $\omega\neq 0\in{\mathbb{Z}}[\omega^{\pm}]/J$. Then, ${\mathbb{Z}}[\omega^{\pm}]/J$ is a quandle under the operation

[TABLE]

for any $x,y\in{\mathbb{Z}}[\omega^{\pm}]/J$. Such a quandle is referred to as an Alexander quandle. When $\omega=1$, that is, $x\ast y=x$, we refer to the quandle as a trivial quandle.

Let $D$ be a diagram of the oriented link $L$ and $\mathcal{A}(D)$ be the set of arcs of $D$. The map $C:\mathcal{A}(D)\to X$ is referred to as an $X$-coloring of $D$ [4, Definition.4.1] if the equation $C(\gamma_{1})\ast C(\gamma_{2})=C(\gamma_{3})$ is applicable at every crossing of $D$, where $\gamma_{2}$ is the over arc at a crossing and $\gamma_{1}$ and $\gamma_{3}$ are under arcs such that $\gamma_{1}$ is on the right side of $\gamma_{2}$ (Figure 2). If $X$ is a finite quandle, then the number of $X$-colorings of $D$ is an invariant of link $L$ [7].

We review the quandle cochain complex [3]. Let $(X,\ast)$ be a quandle of finite order and $A$ be an abelian group. Consider an abelian group,

[TABLE]

For $n\geq 1$, we define the coboundary map of the set above, $\delta_{n}:C_{Q}^{n}(X;A)\longrightarrow C_{Q}^{n+1}(X;A)$, as follows:

\displaystyle\delta_{n}(f)(x_{1},\dots,x_{n+1}):=\sum_{i=2}^{n+1}(-1)^{i}\bigl{(}f(x_{1},\dots,x_{i-1},x_{i+1},\dots,x_{n+1})\\ \hskip 120.0pt-f(x_{1}\ast x_{i},\dots,x_{i-1}\ast x_{i},x_{i+1},\dots,x_{n+1})\bigr{)}.

We easily see that $\delta_{n+1}\circ\delta_{n}=0$. The cohomology of this complex $(C_{Q}^{n}(X;A),\delta_{n})$ is denoted by $H_{Q}^{n}(X;A)$ and referred to as the quandle cohomology of $X$ (with coefficient group $A$).

Let $p$ be a prime number, $m$ be a positive integer, and $q=p^{m}$. We consider the cohomology group, $H^{n}_{Q}(X;A)$, of Alexander quandle $X$ on $\mathop{\mathbb{F}_{q}}\nolimits$ with binary operation (1). The cochain complex $(C_{Q}^{n}(X;A),\delta_{n})$ is described according to [12, 13], as follows: We note that for an arbitrarily fixed $(a_{1},\dots,a_{n})\in X^{n}$, the map $g_{a_{1},\dots,a_{n}}:X^{n}\to\mathop{\mathbb{F}_{q}}\nolimits$ defined by

[TABLE]

can be presented by a polynomial of the form

[TABLE]

Hence, by considering a linear sum of such maps, any map from $X^{n}$ to $\mathop{\mathbb{F}_{q}}\nolimits$ can be presented by a polynomial with respect to $x_{1},\dots,x_{n}$. Furthermore, by introducing new variables, $U_{1}=x_{1}-x_{2},\ U_{2}=x_{2}-x_{3},\dots,U_{n-1}=x_{n-1}-x_{n},\ U_{n}=x_{n}$, we identify the cochain group $C_{Q}^{n}(X;\mathop{\mathbb{F}_{q}}\nolimits)$ with

[TABLE]

It is shown in [13] that the coboundary map is presented by

[TABLE]

Theorem 2.1** ([12, Theorem 2.2]).**

We fix $\omega\in\mathop{\mathbb{F}_{q}}\nolimits$ with $\omega\neq 0,1$. Let $X$ be the corresponding Alexander quandle on $\mathop{\mathbb{F}_{q}}\nolimits$. Then, the following set provides a basis of the second cohomology $H_{Q}^{2}(X;\mathop{\mathbb{F}_{q}}\nolimits)$:

[TABLE]

We define the following sets of polynomials to obtain a basis of $H_{Q}^{3}(X;\mathop{\mathbb{F}_{q}}\nolimits)$: We recall that 3-cocycles are represented by polynomials in $U_{1}$, $U_{2}$, and $U_{3}$,

[TABLE]

where

[TABLE]

Further, we consider

[TABLE]

where

[TABLE]

and condition (2) is given by

[TABLE]

Theorem 2.2** ([13, Theorem 2.11]).**

We fix $\omega\in\mathop{\mathbb{F}_{q}}\nolimits$ with $\omega\neq 0,1$. Let X be the Alexander quandle on $\mathop{\mathbb{F}_{q}}\nolimits$ with binary operation $(\ref{10001})$. Then, $I_{1}\cup I_{2}\cup I_{3}\cup I_{4}\cup I_{0}$ gives a basis of $H_{Q}^{3}(X;\mathop{\mathbb{F}_{q}}\nolimits)$.

Remark 2.3**.**

The original paper [13, Theorem 2.11] claimed a certain cocycle, “$\Psi(a,q_{1})$,” as part of its basis. However, this is an error, and the proof was corrected in another paper [11].

Using a quandle 2-cocycle, $f$, we define the weight, $W_{f}(x;C)$, at the crossing $x$ of diagram $D$ with coloring $C$ in $X$ for the two types of crossings as follows.

[TABLE]

[TABLE]

The quandle 2-cocycle invariant of knots and links [3, Definition.4.3 and Theorem.4.4] is the following state sum:

[TABLE]

which is an invariant of link $L$. We consider

[TABLE]

A quandle cocycle invariant of a link is associated with a 3-cocycle (as an analogy of group cohomology), $\phi$, of finite quandle $X$. Let $\mathcal{A}(D)$ be the set of arcs of $D$ and $R(D)$ be the set of regions of the underlying immersed curves of $\mathcal{A}(D)$. The map $C:\mathcal{A}(D)\sqcup R(D)\to X$ is an $X$-shadow coloring of $D$ [Definition.4.3 [4]], as shown in Figure 2 for $C|_{\mathcal{A}(D)}:\mathcal{A}(D)\to X$ and in Figure 3 for $C|_{R(D)}:R(D)\to X$.

Using the quandle 3-cocycle $\phi$, we define the weight, $W_{\phi}(x;C)$, at crossing $x$ of diagram $D$ with shadow coloring $C$ in $X$ for the two types of crossings as follows.

[TABLE]

[TABLE]

The shadow cocycle invariant of knots and links [3, Definition.5.5 and Theorem.5.6] is the following state sum:

[TABLE]

which is an invariant of link $L$. We consider

[TABLE]

Next, we define Vassiliev invariants. Let $\mathcal{K}$ be a vector space over ${\mathbb{C}}$ freely spanned by the isotopy classes of oriented knots in $S^{3}$. A singular knot is an immersion of $S^{1}$ into $S^{3}$, whose singularities are transversal double points. We regard a singular knot as a linear sum in $\mathcal{K}$ obtained by the relation shown in the following figure 4.

Let $B$ be an abelian group, $\mathcal{K}_{d}$ denote the vector subspace of $\mathcal{K}$ spanned by singular knots with $d$ double points. We refer to a linear map, $v:\mathcal{K}\to B$, as a Vassiliev invariant of degree $d$ if $v|_{\mathcal{K}_{d+1}}=0$ [5, 6, 19].

3 Relation between quandle shadow cocycle invariants and Vassiliev invariants

Let $L$ be an oriented link and $b\in B_{n}$ be a braid such that the closure is isotopic to $L$.

Definition 3.1** ([9]).**

The construction of the general quantum invariable is explained below. Let $V$ be a vector space over ${\mathbb{C}}$. We obtain a representation, $\psi_{n}:B_{n}\to End(V^{\otimes n})$, defined by

[TABLE]

Such a map, $\psi_{n}$, given in (3) always satisfies $\psi_{n}(\sigma_{i}\sigma_{j})=\psi_{n}(\sigma_{j}\sigma_{i})\,\,(|i-j|\geq 2)$. To obtain $\psi_{n}(\sigma_{i}\sigma_{i+1}\sigma_{i})=\psi_{n}(\sigma_{i+1}\sigma_{i}\sigma_{i+1})$, The matrix $R$ is required to satisfy the following relation:

[TABLE]

We refer to this equation as the Yang–Baxter equation, and its solution is referred to as an $R$-matrix.

Theorem 3.2** (chap.I [18] and chap.X [10]).**

We regard $R$ as an $R$-matrix and $h\in End(V)$ as a linear map that satisfies

• •

{\rm trace}_{2}\bigl{(}(id_{V}\otimes h)\cdot R^{\pm}\bigr{)}=id_{V},

• •

$R\cdot(h\otimes h)=(h\otimes h)\cdot R$.

Then, a {\rm trace}\bigl{(}h^{\otimes n}\cdot\psi_{n}(b)\bigr{)} is unchanged by Markov move I and II. Therefore, this invariant is an isotopy invariant of $L$.

We refer to these invariants as the isotopy invariants of $L$ obtained from an $R$-matrix. The invariant calls the operator invariant of $L$ associated with the $R$-matrix.

Theorem 3.3** ([15],[17]).**

Let $f$ be a quandle $2$-cocycle. We consider the following $R$-matrix:

[TABLE]

According to Theorem 3.2, we obtain the operator invariant using the $R$-matrix. This operator invariant is equal to a quandle cocycle invariant, $\Phi_{f}(L)$.

Let $p$ be a prime number, $m$ be a positive integer, and $q=p^{m}$; $\mathop{\mathbb{F}_{q}}\nolimits$ is the finite field of order $q=p^{m}$. Let $\chi$ be an element of a integer rings, $a$, $b$ be a positive integers, and $\hbar$ be $2\pi\sqrt{-1}\chi/ap$. We think about Maclaurin expansions by substituting $e^{a\hbar}$ and $e^{b\hbar}$ for $t$ and $\omega$ of quandle cocycle invariants $\Phi_{f}(L)$ or of quandle shadow cocycle invariants $\Phi_{\phi}(L)$ respectively. Let $\Phi_{\phi}(L)|_{t=e^{a\hbar},\omega=e^{b\hbar}}$ be $\sum_{d=0}^{\infty}u_{d}(L)\hbar^{d}$, then $u_{d}(L)$ gained by the value of $\Phi_{\phi}(L)$ are not unique. Furthermore, since ${\mathbb{Z}}[\mathop{\mathbb{F}_{q}}\nolimits]$ is not a complete field, $\Phi_{\phi}(L)|_{t=e^{a\hbar},\omega=e^{b\hbar}}\neq\sum_{d=0}^{\infty}u_{d}(L)\hbar^{d}$.

Theorem 3.4**.**

$(d!u_{d}(L)\bmod p)\in{\mathbb{Z}}/p{\mathbb{Z}}$* gained by the value of $\Phi_{\phi}(L)$ are unique.*

Proof.

For any $n\in{\mathbb{Z}}$, there exist $k\in{\mathbb{Z}}$ and $0\leq i<p$ such that $n=kp+i$.

[TABLE]

∎

Remark 3.5**.**

Owing to the Maclaurin expansions above do not hold, we define a new expansion as a ring homomorphic mapping ${\mathbb{Z}}[\mathop{\mathbb{F}_{q}}\nolimits]\cong{\mathbb{Z}}[t]/(t^{p}-1,g(t))\to{\mathbb{Z}}/p{\mathbb{Z}}[[\hbar]]$ be $e^{a\hbar}\mapsto 0!u_{0}(K)+1!u_{1}(K)\hbar+2!u_{2}(K)\hbar^{2}+\cdots=\sum_{d=0}^{\infty}d!u_{d}(L)\hbar^{d}$, where $g(t)$ is a polynomial satisfying $\mathop{\mathbb{F}_{q}}\nolimits\cong{\mathbb{F}}_{p}[t]/(g(t))$ and $g(e^{a\hbar})=0$, and $u_{d}(L)$ are sums of coefficients of the degree $d$.

Theorem 3.6**.**

Let $A$ be $\mathop{\mathbb{F}_{q}}\nolimits$, $J$ be any ideal of ${\mathbb{Z}}[\omega^{\pm}]$, $\omega$ be a unit of ${\mathbb{Z}}[\omega^{\pm}]/J$. $f\in H^{2}_{Q}({\mathbb{Z}}[\omega^{\pm}]/J;\mathop{\mathbb{F}_{q}}\nolimits)$ be a quandle $2$-cocycle of the Alexander quandle ${\mathbb{Z}}[\omega^{\pm}]/J$ using $\omega$. We substitute $t=e^{a\hbar}$ and $\omega=e^{b\hbar}$ in

[TABLE]

Here, $g(t)$ is the polynomial on the finite field ${\mathbb{F}}_{p}$ and $\mathop{\mathbb{F}_{q}}\nolimits\cong{\mathbb{F}}_{p}[t]/(g(t))$ and $h_{1}(\omega),\ldots,h_{\ell}(\omega)$ are relations of $\omega$ by coloring $L$. Furthermore, there exist positive integer $a$, $b$ such that $g(e^{a\hbar})=h_{1}(e^{b\hbar})=\cdots=h_{\ell}(e^{b\hbar})=0$. We obtain following power series

[TABLE]

Then, $(d!u_{d}(L)\bmod p)\in{\mathbb{Z}}/p{\mathbb{Z}}$ is a Vassiliev invariant of degree $d$ of $L$.

Proof.

Firstly, we show that $d!u_{d}(L)$ is Vassiliev invariant of degree $d$ of $L$. In the construction of quandle cocycle invariants, we associate a $R$-matrix of equation (4) and its inverse $R^{-1}$ to a positive and a negative crossing respectively. According to equation (4), We obtain the following $R$-matrix:

[TABLE]

We associate matrix $R$ and its inverse, $R^{-1}$, to the positive and negative crossings of $D$ respectively. Since $x^{\prime}=y$ and $y^{\prime}=x$ when $\hbar=0$, matrix elements of the $R$-matrix $R^{xy}_{yx}$ are $1$. In addition, matrix elements of the inverse $R$ matrix ${R^{-1}}^{yx}_{xy}$ are $1$. Next, matrix elements change $x$ for $y$ and $x^{\prime}$ for $y^{\prime}$:

[TABLE]

Since $y^{\prime}=x$ and $x^{\prime}=y$ when $\hbar=0$, matrix elements of the $R$-matrix $R^{xy}_{yx}$ are $1$. In addition, matrix elements of the inverse $R$ matrix ${R^{-1}}^{xy}_{yx}$ are $1$. Hence, $R=R^{-1}$ when $\hbar=0$. Therefore, $R-R^{-1}=0$ is a matrix whose entries are divisible by $\hbar$ in ${\mathbb{Z}}/p{\mathbb{Z}}[[\hbar]]$. This difference is associated to the double point of a singular knot that occurs in the definition of a Vassiliev invariant. Therefore, if $L$ is a singular link with exactly $d+1$ singular points, then $\Phi_{f}(L)|_{t=e^{a\hbar},\omega=e^{b\hbar}}\in{\mathbb{Z}}/p{\mathbb{Z}}[[\hbar]]$ is divisible by $\hbar^{d+1}$. Hence, the coefficient of $\hbar^{d}$ is equal to [math] for such singular links.

Secondly, since Theorem 3.4, the value of $(d!u_{d}(L)\bmod p)\in{\mathbb{Z}}/p{\mathbb{Z}}$ are unique for any $d$. ∎

Theorem 3.7**.**

The notation is the same as that used in Theorem 3.6. Let $\phi\in H^{3}_{Q}({\mathbb{Z}}[\omega^{\pm}]/J;\mathop{\mathbb{F}_{q}}\nolimits)$ be a quandle $3$-cocycle of the Alexander quandle ${\mathbb{Z}}[\omega^{\pm}]/J$ using $\omega$. We substitute $t=e^{a\hbar}$ and $\omega=e^{b\hbar}$ in

[TABLE]

Here, $g(t)$ is the polynomial on the finite field ${\mathbb{F}}_{p}$ and $\mathop{\mathbb{F}_{q}}\nolimits\cong{\mathbb{F}}_{p}[t]/(g(t))$ and $h_{1}(\omega),\ldots,h_{\ell}(\omega)$ are relations of $\omega$ by coloring $L$. Furthermore, there exist positive integer $a$, $b$ such that $g(e^{a\hbar})=h_{1}(e^{b\hbar})=\cdots=h_{\ell}(e^{b\hbar})=0$. We obtain following power series

[TABLE]

Then, $(d!u_{d}(L)\bmod p)\in{\mathbb{Z}}/p{\mathbb{Z}}$ is a Vassiliev invariant of degree $d$ of $L$.

Proof.

We obtain the following $R$-matrix:

[TABLE]

We associate matrix $R$ and its inverse, $R^{-1}$, to the positive and negative crossings of $D$, respectively. These matrices coincide when $\hbar=0$. By definition (ii) of a quandle, there exists unique $z^{\prime}\in{\mathbb{Z}}[\omega^{\pm}]/J$ in the unbounded region of $D$ such that $R=R^{-1}$. Further proof can be obtained in the same manner as that for Theorem 3.6. ∎

Remark 3.8**.**

According to the above proof, the Vassiliev invariant of $L$ is derived without considering how to select an ideal, $J$, of ${\mathbb{Z}}[\omega^{\pm}]$. Hence, there exist as many Vassiliev invariants of $L$ as there are ideals of ${\mathbb{Z}}[\omega^{\pm}]$. Generally, the finite invariants (Vassiliev invariants) of classical knots are not compatible with the quandle cocycle invariants. When the number of crossings for the knots that are closures (or plat closures) of braids composed of $l$ strings is increased, the values of quandle cocycle invariants are bounded, while the values of Vassiliev invariants increase by the order of the polynomial functions of the crossings. However, the Vassiliev invariants deduced from the shadow cocycle invariants can be defined independently of the manner in which the ideals of the Alexander quandles are selected. Therefore, the values of these Vassiliev invariants are not trivial because the same number of (infinite) ideals is deduced from a shadow cocycle invariant.

Remark 3.9**.**

When $J\neq(0)$, we know that that the cochain map $f\in C_{Q}^{n}({\mathbb{Z}}[\omega^{\pm}]/J;{\mathbb{Z}})$ does not exist in the injective map $f:{\mathbb{Z}}[\omega^{\pm}]/J\to{\mathbb{Z}}[[\hbar]]$. When $J=(p)$, refer Example 3.10 and when $J=(p,\omega^{2}+\omega+1)$, refer Example 3.11.

Let $p$ be an odd prime number and $J$ be a $(p)$ of the ideal of ${\mathbb{Z}}[\omega^{\pm}]$. We know that $H_{Q}^{3}({\mathbb{Z}}[\omega^{\pm}]/J;{\mathbb{Z}})\cong{\mathbb{Z}}/p{\mathbb{Z}}\,\cite[cite]{[\@@bibref{}{14}{}{}]}$. Hence, we need Theorem3.4 to deduce the Vassiliev invariants from quandle shadow cocycle invariants by using the quandle cocycle of Theorem2.2.

Example 3.10**.**

Let $J$ be a $(3)$ of an ideal of ${\mathbb{Z}}[\omega^{\pm}]$; $\chi$ be an indeterminate of ${\mathbb{Z}}$; and $\hbar$ be $\pi\sqrt{-1}\chi/3$. In addition, we consider $L$ $(2,n)$-torus links and $\phi(x,y,z)$ in the following equation, in which we substitute $p=3$ for the Mochizuki $3$-cocycle [12, 13]:

[TABLE]

Hence, $H_{Q}^{3}(X;{\mathbb{Z}}/3{\mathbb{Z}})\cong{\mathbb{Z}}/3{\mathbb{Z}}$. If $n\equiv 0\pmod{3}$, then

[TABLE]

We substitute $t=e^{2\hbar}$ and $\omega=e^{\hbar}$ in

[TABLE]

[TABLE]

Since Theorem 3.4, $9\equiv 0\pmod{3}$ is the Vassiliev invariant of degree [math]. $n/3\pmod{3}$ is the Vassiliev invariant of degree $2$.

Example 3.11**.**

Let $J$ be a $(2,\omega^{2}+\omega+1)$ of an ideal of ${\mathbb{Z}}[\omega^{\pm}]$; $\chi$ be an indeterminate of ${\mathbb{Z}}$; and $\hbar$ be $\pi\sqrt{-1}\chi/3$. In addition, we consider $L$ $(2,n)$-torus links and $f(x,y)$ in the following equation, in which we substitute $p=2$ and $q=p^{2}$ for the $2$-cocycle [12, 13]:

[TABLE]

Hence, H_{Q}^{2}(X;\mathop{\mathbb{F}_{q}}\nolimits)\cong\big{\{}(x-y)^{2^{v}}y^{2^{u}}\ \big{|}\ \ 0\leq v<u<2\big{\}} by Theorem 2.1. If $n\equiv 0\pmod{3}$, then

[TABLE]

We substitute $t=e^{3\hbar}$ and $\omega=e^{2\hbar}$ in

[TABLE]

[TABLE]

Since Theorem 3.4, $16\equiv 0\pmod{2}$ is the Vassiliev invariant of degree [math]. $2n\equiv 0\pmod{2}$ is the Vassiliev invariant of degree $2$.

We obtain the $2$-twist-spun knots [2] of the $(2,n)$-torus knots of the Vassiliev invariants through expansion (6). Thus, we consider the following problem of surface $2$-knots:

Problem 3.12**.**

$(1)$ Do singular curves corresponding to singular points exist ?

$(2)$ Can quantum invariants be defined for 2-knots ?