Twisted cocycle invariants

In [CES02], a cocycle invariant using twisted quandle cohomology theory was defined. Let $X$ be a quandle, and $A$ be an Alexander quandle with variable $t$ (typically $A=\mathbb{Z}_p[t, t^{-1}]/h(t)$ for a prime $p$ and a polynomial $h(t)$). A function $\phi : X \times X \rightarrow A$ is called a twisted quandle $2$-cocycle if it satisfies the twisted $2$-cocycle condition

$$t [ \phi(y,z) - \phi(x,z)+ \phi(x,y) ]- [\phi(y,z) - \phi(x*y, z) + \phi(x*z, y*z)]=0,$$

for all $x,y,z \in X$, and $\phi(x,x)=0, \forall x \in X$.

Let $K$ be an oriented knot diagram with orientation normal vectors. The underlying projection of the diagram divides the $3$-space into regions. For a crossing $\tau$, among four regions adjacent to $\tau$, there is a unique region $R$ such that all normal vectors of arcs near $\tau$ point from $R$ to other regions. Such a region $R$ is called the source region of $\tau$. Take an oriented arc $\ell$ from the region at infinity to a region $H$ such that $\ell$ intersects the arcs (missing crossing points) of the diagram transversely in finitely many points. The Alexander numbering ${\cal L}(H)$ of a region $H$ is the number of such intersections counted with signs. If the orientation of $\ell$ agrees with the normal vector of the arc where $\ell$ intersects, then the intersection is positive (counted as $1$), otherwise negative (counted as $-1$). The Alexander numbering ${\cal L}(\tau)$ of a crossing $\tau$ is defined to be ${\cal L}(R)$ where $R$ is the source region of $\tau$.

For a coloring ${\cal C}$ of $K$ by $X$, the twisted $($Boltzmann$)$ weight at a triple point $\tau$ is defined by $B_T(\tau, {\cal C})= [\phi(x,y)^{\epsilon (\tau)} ]^{t^{-{\cal L}(\tau)}} ,$ where $\phi$ is a twisted $2$-cocycle, and $\epsilon(\tau)$ is the sign of the crossing $\tau$. This is in multiplicative notation, so that the action of $t$ on $a \in A$ is written by $a^t$. The colors $x,y$ assigned near $\tau$ are chosen in the same manner as in the ordinary $2$-cocycle invariant, and ${\cal L}(\tau)$ is the Alexander numbering of $\tau$. The twisted quandle $2$-cocycle invariant is the state-sum

$$\Phi_T (K) = \sum_{{\cal C}} \prod_{\tau} B_T( \tau, {\cal C}).$$

The value of the weight $B_T( \tau, {\cal C})$ is in the coefficient group $A$ written multiplicatively, and the value of the state-sum is again in the group ring ${\bf Z}[A]$. It was proved [CES02] by checking Reidemeister moves that $\Phi_T (K)$ is an invariant of knots.