Quandle $3$-Cocycle Invariants of Classical Knots

Let $X$ be a finite quandle and $A$ an abelian group. A function $\theta : X \times X \times X \rightarrow A$ is called a quandle $3$-cocycle if it satisfies the $3$-cocycle condition

$${\theta (x,z,w)- \theta(x,y,w)+\theta(x,y,z) -\theta(x*y,z,w)}$$

$$+\theta(x*z,y*z,w)-\theta(x*w,y*w,z*w)=0,$$

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

Let $\tilde{{\mathcal C}}\in {\rm Colr}_X(D)$ be a coloring of arcs and regions of a given diagram $D$ of a classical knot $K$. Let $(x,y,z) (=(x_{\tau}, y_{\tau}, z_{\tau}))$ be the ordered triple of colors at a crossing $\tau$, see Fig.  of Chapter $2$. Let $\theta$ be a $3$-cocycle. Then the weight in this case is defined by $B( {\mathcal C}, \tau)=\theta(x_{\tau}, y_{\tau}, z_{\tau})^{\epsilon(\tau)}$ where $\epsilon(\tau)$ is $\pm 1$ for positive and negative crossing, respectively. Then the ($3$-)cocycle invariant is defined in a similar way to $2$-cocycle invariants by $\Phi_{\theta}(K) =\{ \sum_{\tau} B( {\mathcal C}, \tau ) \vert {\mathcal C}\in {\rm Colr_X(D)} \}$. The multiset version is defined similarly.

Example 4.3   Let $D$ be the closure of $\sigma_1^3$, a diagram of a trefoil. Take $X=R_3$ for a quandle and let $\theta(x,y,z)=(x-y)(y^2+yz+z^2)z \pmod{3}$ for $x, y, z \in R_3=\mathbb{Z}_3$. (This formula came from $\theta (x,y,z)= (x-y) [(2 z^p - y^p) - (2z-y)^p ]/p \pmod{p}$ for $p=3$ that will be discussed later.) If the source region of all three crossings is colored by 0, the top left and right arcs are colored by $(1,2)$, respectively, then the contribution is

$$\theta(0,1,2) + \theta(0,2,0) + \theta(0,0,1)=(-1)(1+2+1)(2) + 0 + 0 = 1 \quad {\rm in} \quad \mathbb{Z}_3,$$

and by computing all colors, we obtain $\Phi_{\theta}(K)=9 + 18u$.