Quandle Cocycle Invariants with Actions on Coefficients

In [CES02], a cocycle invariant using twisted quandle cohomology theory was defined. Let $X$ be a quandle, and $A$ be an Alexander quandle with a 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 $\tau$, let $R$ be the source region (i.e., a unique region among four regions adjacent to $\tau$ such that all normal vectors of arcs near $\tau$ point from $R$ to other regions). Take an oriented arc $\ell$ from the region at infinity to a given region $H$ such that $\ell$ intersects the arcs (missing crossing points) of the diagram transversely in finitely many points. The Alexander numbering ${\mathcal L}(H)$ 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 ${\mathcal L}(\tau)$ of a crossing $\tau$ is defined to be ${\mathcal L}(R)$ where $R$ is the source region of $\tau$.

For a coloring ${\mathcal C}$ of a diagram $D$ od a classical knot $K$ by $X$, the twisted $($Boltzmann$)$ weight at $\tau$ is defined by $B_T(\tau, {\mathcal C})= [\phi(x_{\tau},y_{\tau})^{\epsilon (\tau)} ]^{t^{-{\mathcal L}(\tau)}} ,$ 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 $x_{\tau},y_{\tau}$ assigned near $\tau$ are chosen in the same manner as in the ordinary $2$-cocycle invariant (source colors, or ordered pair of colors at $\tau$) and ${\mathcal L}(\tau)$ is the Alexander numbering of $\tau$. The twisted quandle $2$-cocycle invariant is the state-sum

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

The value of the weight $B_T( \tau, {\mathcal C})$ 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.