R-Matrix Invariants

KnotTheory` can compute knot and link invariants associated with matrix solutions of the Yang-Baxter equation, using the program REngine: (For In[1] see Setup)

In more detail, let $V$ be a free module of finite rank over a commutative ring, and let $R,{\bar {R}}\in End(V\otimes V)$ be invertible. Further, let there be invertible endomorphisms $M_{\leftarrow },M_{\rightarrow },M^{\leftarrow },M^{\rightarrow }\in End(V)$; these correspond to McupL, McupR, McapL, and McapR respectively. Fixing a basis of V, we can regard all of these endomorphisms as matrices.

For an oriented knot or link $K$, REngine returns the product $\tau (K)$, which is computed as follows:

1. Find a Morse link presentation of $K$ such that it is composed only of crossings of the X[k, Over/Under, Down, Down] variety; this is easily accomplished by adding caps and cups where necessary. Take the chosen axis to be vertical, pointing upwards, so cups and caps are maxima and minima with respect to the vertical height function.
2. Label the edges in between events in the Morse link presentation of K.
3. Associate each event with one of the matrices as shown:

   • X[k, Over, Down, Down]:$\Rightarrow R_{ac}^{bd}$

     X[k, Under, Down, Down]: $\Rightarrow {\bar {R}}_{ac}^{bd}$