$2$-Cocycle invariants

The cocycle invariant for classical knots [CJKLS03] using quandle $2$-cocycles was defined as follows. Let $X$ be a finite quandle and $A$ an abelian group, typically a finite cyclic group $\mathbb{Z}_n$, $n \in \mathbb{Z}$, $n>1$.

A function $\phi : X \times X \rightarrow A$ is called a quandle $2$-cocycle if it satisfies the $2$-cocycle condition

$$\phi(x,y) - \phi(x,z) + \phi(x*y, z) - \phi(x*z, y*z)=0, \quad \forall x,y,z \in X$$

and $\phi(x,x)=0, \forall x \in X$. Let ${\cal C}$ be a coloring of a given knot diagram $K$ by $X$. The Boltzmann weight $B( {\cal C}, \tau)$ at a crossing $\tau$ of $K$ is then defined by $B( {\cal C}, \tau)=\phi(x_{\tau}, y_{\tau})^{\epsilon(\tau)}$, where $(x_{\tau}$, $y_{\tau})$ is the ordered pair of colors at $\tau$ and $\epsilon(\tau)$ is the sign ($\pm 1$) of $\tau$. Here $B( {\cal C}, \tau)$ is an element of $A$ written multiplicatively. The formal sum (called a state-sum) in the group ring $\mathbb{Z}[A]$

$$\Phi_{\phi} (K) = \sum_{{\cal C}\in {\rm Col_X(K)}} \prod_{\tau} B( {\cal C}, \tau )$$

is called the quandle $2$-cocycle invariant. A value (an element of $\mathbb{Z}[A]$) is written as $\sum_{g \in G} a_g g$, so that if $A=\mathbb{Z}_n=\{ u^k \vert 0 \leq k < n \}$, a value is written as a polynomial $\sum_{k=0}^{n-1} a_k u^k$. That $\Phi_{\kappa} (K)$ is a knot invariant is proved easily by checking Reidemeister moves [CJKLS03].

The cocycle invariant can be also written as a family (multi-set, a set with repetition allowed) of weight sums [Lop03]

$$\left\{ \left. \sum_{\tau} B( {\cal C}, \tau ) \right\vert {\cal C}\in {\rm Col_X(K)} \right\}$$

where now the values of $B$ in $A$ are denoted by additive notation. Then this is well-defined for any quandle of any cardinality.