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 [math]\displaystyle{ V }[/math] be a free module of finite rank over a commutative ring, and let [math]\displaystyle{ R, \bar{R} \in End(V \otimes V) }[/math] be invertible. Further, let there be invertible endomorphisms [math]\displaystyle{ M_\leftarrow, M_\rightarrow, M^\leftarrow, M^\rightarrow \in End(V) }[/math]; 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 [math]\displaystyle{ n = dim(V) }[/math], [math]\displaystyle{ R_{ab}^{cd} }[/math] refers to the element in row [math]\displaystyle{ n(c-1)+d }[/math] row and column [math]\displaystyle{ n(a-1) + b }[/math] column.
For an oriented knot or link [math]\displaystyle{ K }[/math], REngine returns the product [math]\displaystyle{ \tau(K) }[/math], which is computed as follows:
- Find a Morse link presentation of [math]\displaystyle{ K }[/math] 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. - Label the edges in between events in the Morse link presentation of [math]\displaystyle{ K }[/math].
- Associate each event with one of the matrices as shown, with the indices given by the labelling:
X[k, Over, Down, Down]:
[math]\displaystyle{ \Rightarrow R_{ab}^{cd} }[/math]X[k, Under, Down, Down]:
[math]\displaystyle{ \Rightarrow\bar{R}_{ab}^{cd} }[/math]
- Define [math]\displaystyle{ \tau(K) }[/math] 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 [math]\displaystyle{ n=dim(V) }[/math].
As an example, let K be the left-handed trefoil, as shown. Then
[math]\displaystyle{ \tau(K) = \sum {M_\leftarrow}_{ab} }[/math]
