R-Matrix Invariants

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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.

The following notation is useful: for $n=dim(V)$, $R_{ab}^{cd}$ refers to the element of R in row $n(c-1)+d$ row and column $n(a-1)+b$.

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 accomplished by adding caps and cups where necessary.
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, with the indices given by the labelling:
   • X[k, Over, Down, Down]:$\Rightarrow R_{ab}^{cd}$
   • X[k, Under, Down, Down]: $\Rightarrow {\bar {R}}_{ab}^{cd}$
4. Define $\tau (K)$ as the result of taking the product of the matrices associated with the elements of K, and summing over repeated indices; each sum runs from 1 to $n=dim(V)$.

A trefoil knot

As an example, let K be the trefoil as shown, with the chosen axis upwards, and the strands going down through all the crossings. Then

$\tau (K)=\sum {M_{\leftarrow }}_{ab}{M_{\rightarrow }}_{cd}R_{bc}^{ef}R_{ef}^{gh}R_{gh}^{ij}{M^{\rightarrow }}_{ai}{M^{\leftarrow }}_{jd}$

where the summation is carried out on all the indices a through i, each ranging from 1 to n.

Determining that $\tau (K)$ is actually an invariant is simply a matter of checking the equality of the outcomes of Reidemeister moves in various configurations and orientations, as well as a few 'topological' moves. Note that REngine does not perform any checks; the utility TestRMatrix does the tests required to determine regular isotopy.