Quandles, colorings, and cocycle invariants

Again we refer the reader to the article posted at http://shell.cas.usf.edu/quandle under the title Background, for the following terms we use: quandles, Alexander quandles, colorings of knot diagrams, coloring of regions of a knot diagram, quandle $2$- and $3$-cocycles, $2$- and $3$-cocycle invariants of knots.

Here we review a definition of cocycle invariants in terms of multisets, that will be used in this article.

Let $K$ be a knot diagram on the plane. Let $X$ be a finite quandle and $A$ an abelian group. Let $\phi : X \times X \rightarrow A$ be a quandle $2$-cocycle, which can be regarded as a function satisfying 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 $B( {\cal C}, \tau)=B_{\phi}( {\cal C}, \tau)$ at a crossing $\tau$ of $K$ is then defined by $B( {\cal C}, \tau)=\epsilon(\tau) \phi(x_{\tau}, y_{\tau})$, where $(x_{\tau}$, $y_{\tau})$ is the ordered pair of colors at $\tau$ and $\epsilon(\tau)$ is the sign ($\pm 1$) of $\tau$. Then the $2$-cocycle invariant $\Phi(K)=\Phi_{\phi}(K)$ in a multiset form is defined by

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

where ${\rm Col_X(K)}$ denotes the set of colorings of $K$ by $X$. (A multiset a collection of elements where a single element can be repeated multiple times, such as $\{ 0, 0, 1, 1, 1 \}$).

Let $\theta : X \times X \times X \rightarrow A$ be a quandle $3$-cocycle, which can be regarded as a function satisfying

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

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

and $\theta(x,x,y)=0=\theta(x,y,y), \forall x, y \in X$.

Let ${\cal C}$ be a coloring of arcs and regions of a given diagram $K$. Let $(x,y,z) (=(x_{\tau}, y_{\tau}, z_{\tau}))$ be the ordered triple of colors at a crossing $\tau$. Then the weight in this case is defined by $B( {\cal C}, \tau)=\epsilon(\tau) \phi(x_{\tau}, y_{\tau}, z_{\tau}).$ The ($3$-)cocycle invariant is defined in a similar way to $2$-cocycle invariants by the multiset $\Phi_{\theta}(K) =\{ \sum_{\tau} B( {\cal C}, \tau ) \ \vert \ {\cal C}\in {\rm Col_X(K)} \}$, where ${\rm Col_X(K)}$ denotes the set of colorings with region colors of $K$ by $X$.