Quandle $3$-Cocycle Invariants of Knotted Surfaces

For a fixed finite quandle $X$ and a $3$-cocycle $\theta$, we define a knotted surface invariant as follows: Let $K$ be a knotted surface diagram and let ${\mathcal C} : {\mathcal R} \rightarrow X$ be a coloring of $K$, where ${\mathcal R}$ is the set of sheets of $K$. The Boltzman weight at a triple point $\tau$ is defined by $B(\tau, {\mathcal C}) = \theta(x_{\tau},y_{\tau},z_{\tau})^{\epsilon (\tau)},$ where $\epsilon(\tau)$ is the sign of the triple point $\tau$. (A triple point is positive if and only if the normal vectors of the top, middle, and bottom sheets, in this order, agree with the (right-hand) orientation of $\mathbb{R}^3$ [CrSt98]. In the right of Fig.  of Chapter $2$, the triple point $\tau$ is positive.) The colors $x_{\tau},y_{\tau},z_{\tau}$ are the colors of the bottom, middle, and top sheets, respectively, around the source region of $\tau$. The source region is the region from which normals of top, middle and bottom sheets point. The cocycle invariant is defined by

$$\Phi(K)= \sum_{{\mathcal C}} \prod_{\tau} B( \tau, {\mathcal C})$$

as before. It was shown in [CJKLS03] that $\Phi(K)$ is an invariant for knotted surfaces, called the $($quandle$)$ cocycle invariant. The multiset form is similarly defined.

Since it requires some preliminary expositions to explain examples of how to compute this invariant for surfaces, we refer the reader to other publications, such as [AS03*,SatShi01b*,SatShi04], for example.