4.1 Biquandle 2-cocycle invariants

An important extension of quandle coloring invariants comes from the (co)homology theory of quandles. This kind of (co)homology theory was first introduced for racks in [Reference Fenn and Rourke8] and later a modified version was introduced for quandles in [Reference Carter, Jelsovsky, Kamada, Langford and Saito1]. Roughly speaking, for a given quandle 2-cocycle and a fixed quandle coloring, one can assign a Boltzmann weight to each crossing point. Taking the product of Boltzmann weights for all crossing points, then the sum of all these products among all the colorings gives rise to a state-sum invariant for knots, called the quandle cocycle invariant. The quandle coloring invariants can be recovered from the quandle cocycle invariants if the quandle 2-cocycle is a coboundary. For biquandles, Carter, Elhamdadi, and Saito introduced a homology theory for the set-theoretic Yang–Baxter equations in [Reference Carter, Elhamdadi and Saito2]. The state-sum invariants can be similarly defined by using biquandle cocycles.

Now we take a quick review of the definition of biquandle cohomology groups. For a given biquandle X, let $C_n^{BR}(X)$ be the free abelian group generated by n-tuples $(x_1, \cdots , x_n)$ , where $x_i\in X$ . We define $d_n(x_1, \cdots , x_n)=$

$$ \begin{align*} \sum\limits_{i=1}^n(-1)^i((x_1, \ldots, x_{i-1}, x_{i+1}, \ldots, x_n)-(x_1\ast x_i, \ldots, x_{i-1}\ast x_i, x_{i+1}\circ x_i, \ldots, x_n\circ x_i)) \end{align*} $$

and extend it linearly to obtain a map $d_n: C_n^{BR}(X)\to C_{n-1}^{BR}(X)$ . In particular, when $n\leq 1$ we set $d_n=0$ . It is a routine exercise to verify that $d_{n-1}\circ d_n=0$ , thus $C_{\ast }^{BR}(X)=\{C_n^{BR}(X), d_n\}$ is a chain complex and the homology groups are called the birack homology groups.

In order to define the biquandle homology groups, we need to consider the chain complex $C_{\ast }^{BQ}(X)=\{C_n^{BQ}(X), d_n\}$ , where $C_n^{BQ}(X)=C_n^{BR}(X)/C_n^{BD}(X)$ . Here $C_n^{BD}(X)$ denotes the free abelian group generated by n-tuples $(x_1, \ldots , x_n)$ where ${x_i=x_{i+1}}$ for some $1\leq i\leq n-1$ . It turns out that $C_{\ast }^{BD}(X)=\{C_n^{BD}(X), d_n\}$ is a subcomplex of $C_{\ast }^{BR}(X)$ and the nth biquandle homology group and nth biquandle cohomology group are defined as

$$ \begin{align*} H_n^{BQ}(X)=H_n(C_{\ast}^{BQ}(X))\text{ and }H_{BQ}^n(X)=H^n(C_{BQ}^{\ast}(X)), \end{align*} $$

where $C_{BQ}^{\ast }(X)=\text {Hom}(C_{\ast }^{BQ}(X), \mathbb {Z})$ . For an abelian group A, the homology $H_n^{BQ}(X; A)$ and cohomology $H_{BQ}^n(X; A)$ with coefficient group A can be similarly defined.

Let X be a finite biquandle, A an abelian group, $\phi $ a biquandle 2-cocycle and $f\in $ Aut $(X)$ an automorphism of X satisfying $x\ast y=x\circ f(y)$ for any $x, y\in X$ . For a link L in $\Sigma _g\times S^1$ , choose a link diagram D of it. For a fixed X-coloring $\rho $ of D, consider a crossing points c as in Figure 1 and assign a Boltzmann weight ${W_{\phi }^{\rho }(c)=\phi (x, y)^{w(c)}\in A}$ to it. Here $w(c)$ denotes the sign of c. We associate an element $\Phi _{\phi }^f(D)$ of the group ring $\mathbb {Z}[A]$ to D as follows

$$ \begin{align*} \Phi_{\phi}^f(D)=\sum\limits_{\rho}\prod\limits_cW_{\phi}^{\rho}(c)\in\mathbb{Z}[A]. \end{align*} $$

The following theorem suggests that for some suitable choice of $\phi $ , $\Phi _{\phi }^f(D)$ does not depend on the choice of D. Hence it defines a link invariant for links in $\Sigma _g\times S^1$ , we call it the biquandle cocycle invariant of L with respect to f and $\phi $ and denote it by $\Phi _{\phi }^f(L)$ .

Obviously, if one sets $\phi (x, y)=1$ for any $x, y\in X$ , then $\Phi _{\phi }^f(K)$ reduces to the coloring invariant Col $_{(X, f)}(K)$ .

Example 4.2 Consider the two links $L_1, L_2$ , the biquandle X and the automorphism $f\in \text {Aut}(X)$ given in Example 3.2, let us choose an infinite cyclic group A generated by a and a biquandle 2-cocycle $\phi $ defined by

$$ \begin{align*} \phi(1, 1)=\phi(2, 2)=\phi(3, 3)=\phi(1, 2)=\phi(2, 1)=1, \end{align*} $$

and

$$ \begin{align*} \phi(1, 3)=\phi(2, 3)=a, \phi(3, 2)=\phi(3, 1)=a^{-1}. \end{align*} $$

It is easy to see that $\phi $ satisfies the extra condition required in Theorem 4.1. Direct calculation shows that

$$ \begin{align*} \Phi_{\phi}^f(L_1)=2a+2a^{-1}+1\text{ and }\Phi_{\phi}^f(L_2)=5. \end{align*} $$

Comparing with the result given in Example 3.2, we find that the biquandle cocycle invariants are stronger than the biquandle coloring invariants.

For classical knots or virtual knots, it is well known that if the 2-cocycle $\phi $ is a coboundary, then the cocycle invariant reduces to the coloring invariant. In our case, if $\phi =\delta _1\psi $ for some $\psi \in C^1_{BQ}(X)$ , then

$$ \begin{align*} \phi(x, y)=\delta_1\psi(x, y)=\psi d_2(x, y)=\psi(y)^{-1}\psi(y\circ x)\psi(x)\psi(x\ast y)^{-1}. \end{align*} $$

For a fixed coloring $\rho $ , the Boltzmann weight of each crossing point splits into four terms assigned to the four semi-arcs around it. If a semi-arc contains no arrow, then the contributions received by this semi-arc from its two ends cancel out. Hence if the arrowed diagram actually contains no arrow, the product $\prod \limits _cW_{\phi }^{\rho }(c)=1$ . If the arrowed diagram contains some arrows, notice that

$$ \begin{align*} 1=\phi(x, y)\phi(y, f^{-1}(x))=\delta_1\psi(x, y)\delta_1\psi(y, f^{-1}(x)), \end{align*} $$

which implies

$$ \begin{align*} 1\kern1.4pt{=}\kern1.4pt \psi(y)^{-1}\psi(y\circ x)\psi(x)\psi(x\kern1.4pt{\ast}\kern1.4pt y)^{-1}\psi(f^{-1}(x))^{-1}\psi(f^{-1}(x)\circ y)\psi(y)\psi(y\kern1.4pt{\ast}\kern1.4pt f^{-1}(x))^{-1}. \end{align*} $$

Together with the assumption that $y\ast f^{-1}(x)=y\circ x$ , it follows that

$$ \begin{align*} \psi(x)\psi(x\ast y)^{-1}\psi(f^{-1}(x))^{-1}\psi(f^{-1}(x)\circ y)=1. \end{align*} $$

Or equivalently,

$$ \begin{align*} \psi(x)\psi(f^{-1}(x))^{-1}=\psi(x\ast y)\psi(f^{-1}(x)\circ y)^{-1}. \end{align*} $$

Since the elements $x, y$ can be arbitrarily chosen and f is a bijection, the equality above can be rewritten as

$$ \begin{align*} \psi(f(x))\psi(x)^{-1}=\psi(f(x)\ast f(y))\psi(x\circ f(y))^{-1}=\psi(f(x\ast y))\psi(x\ast y)^{-1}, \end{align*} $$

or

$$ \begin{align*} \psi(f(x))\psi(x)^{-1}=\psi(f(x)\ast y)\psi(x\circ y)^{-1}=\psi(f(x\circ y))\psi(x\circ y)^{-1}. \end{align*} $$

Similar to the notion of Latin quandle, we say a biquandle X is Latin if for any $x\in X$ , the maps $x\ast \bullet : X\to X$ and $x\circ \bullet : X\to X$ are both bijective. If either one of these two maps is bijective, then we say X is semi-Latin. Now assume we are given a finite semi-Latin biquandle X, then the two equalities above suggest that for any $x, y\in X$ , we have

$$ \begin{align*} \psi(f(x))\psi(x)^{-1}=\psi(f(y))\psi(y)^{-1}. \end{align*} $$

That is to say, for any $x\in X$ , $\psi (f(x))\psi (x)^{-1}$ is a constant element of the abelian group A. Let us use a to denote it. Similar to Example 3.3, we use $p: \Sigma _g\times S^1\to S^1$ to denote the projection from $\Sigma _g\times S^1$ to $S^1$ and $[\alpha ]\in H_1(S^1)$ be the generator of $H_1(S^1)$ . For an oriented link L in $\Sigma _g\times S^1$ , if $p_{\ast }([L])=k[\alpha ]$ and the 2-cocycle $\phi $ is a coboundary, then we have $\Phi _{\phi }^f(L)=\sum \limits _{\rho }a^k$ . When $k=0$ , this recovers the corresponding result of classical knots.

4.2 Index type invariants

In this subsection, we discuss another application of the quandle coloring invariants. Roughly speaking, what we want to do is to introduce an enhanced version of the writhe of a link diagram. Recall that for a given link diagram D, the writhe of D is the sum of the signs among all the crossing point. This is not a link invariant in general, since it is not preserved under the first Reidemeister move. In order to overcome this, we can assign an index (e.g., an integer, a polynomial or an element of a group, etc) to each crossing point of D and the writhe now turns into a weighted sum. The requirements of the index that make this weighted sum becomes a link invariant were first introduced in [Reference Cheng3] for virtual knots. Here we present a modified version as follows.

Index type invariants of virtual knots have been intensively studied during the past ten years. The readers are referred to [Reference Cheng, Fedoseev, Gao, Manturov and Xu4] for a survey of some recent progress on this topic. In particular, a family of integer-valued index type invariants for knots in $\Sigma _g\times [0, 1]$ was recently introduced in [Reference Cheng, Gao and Xu5]. Now we discuss how to define index type invariants for knots in $\Sigma _g\times S^1$ .

Let L be link in $\Sigma _g\times S^1$ and D a link diagram of it. Assume we are given a finite biquandle X and an automorphism $f\in $ Aut $(X)$ satisfying $x\ast y=x\circ f(y)$ for all $x, y\in X$ , a concrete example of index can be constructed as follows. Consider the group

$$ \begin{align*} &G_X=<(x, y)\in X\times X|(x, x)=(x, y)(y, f^{-1}(x))=1, (x, y)=(x\ast z, y\ast z), (y, z)=\\&\quad(y\circ x, z\circ x), (x, z)=(x\ast y, z\circ y)>. \end{align*} $$

Note that in $G_X$ we also have

$$ \begin{align*} &(x\circ y, z\ast y)=((x\circ y)\ast^{-1}y, z)=((x\circ y)\ast^{-1}y, (z\circ y)\circ^{-1}y)\\&\quad=(x\circ y, z\circ y)=(x, z). \end{align*} $$

On the other hand, for a given abelian group A, any homomorphism $\phi $ from $G_X$ to A satisfies the biquandle 2-cocycle condition mentioned in Theorem 4.1. Furthermore, for any such biquandle 2-cocycle $\phi $ , not only the product of the Boltzmann weights of the three crossings involved in the third Reidemeister move is preserved, actually, now the Boltzmann weight of each crossing is preserved under the third Reidemeister move.

For a fixed coloring of D by X, say $\rho $ , we associate a weight $w_{\rho }(c)=(x, y)\in G_X$ to each of the two crossing points in Figure 1. Now we define the index (associated with X and f) of a crossing point c to be Ind $(c)=\sum _{\rho }w_{\rho }(c)\in \mathbb {Z}G_X$ . In particular, if there exists a homomorphism $\phi $ from $G_X$ to an abelian group A, then we obtain an induced index $\phi (\sum _{\rho }w_{\rho })\in \mathbb {Z}A$ . Now for any element $g\in \mathbb {Z}G_X$ , we define

$$ \begin{align*} a_g(D)= \begin{cases} \sum\limits_{\text{Ind}(c)=g}w(c),& \text{if }g\neq\sum\limits_{\rho}1;\\ \sum\limits_{\text{Ind}(c)=g}w(c)-w(D),& \text{if }g=\sum\limits_{\rho}1. \end{cases} \end{align*} $$

Here $w(D)$ denotes the writhe of D.

Example 4.5 Consider the two links $L_1, L_2$ discussed in Example 3.2, the biquandle X and the automorphism f used in Example 4.2. It is not difficult to find that $G_X=\mathbb {Z}$ , which is generated by $a=(1, 3)=(2, 3)$ . Then we have

$$ \begin{align*} a_g(L_1)= \begin{cases} -1,& \text{if }g=2a+2a^{-1}+1;\\ 0,& \text{otherwise}, \end{cases} \end{align*} $$

and $a_g(L_2)=0$ for any $g\in \mathbb {Z}G_X$ .

Example 4.6 Consider the 2-component link $L_3\subset S^2\times S^1$ depicted in Figure 8. We still use the same biquandle X and automorphism f as that in Example 4.2. Now direct calculation shows that

$$ \begin{align*} a_g(L_3)= \begin{cases} -1,& \text{if }g=1+2a;\\ -1,& \text{if }g=1+2a^{-1};\\ 0,& \text{otherwise}. \end{cases} \end{align*} $$

We would like to remark that if we choose the biquandle 2-cocycle $\phi $ in Example 4.2, the biquandle cocycle invariant $\Phi _{\phi }^f(L_3)$ equals 3, which contains no extra information comparing with the biquandle coloring invariant.

Figure 8

A 2-component link $L_3\subset S^2\times S^1$ .