Knot invariants and the Birman-Wenzl-Murakami algebra

In a previous post I mentioned that the Birman-Wenzl-Murakami algebra leads to link invariants – in this post I’ll detail this moreso.

What is the Birman-Wenzl-Murakami algebra? It is an associative algebra over a field (let’s take it to be the complex field C) depending on two parameters r, q. You can take r and q to be indeterminates and work over the field C(r,q), or you can fix these parameters to have specific values. It doesn’t matter too much yet.

The Birman-Wenzl-Murakami algebra BWn(r,q) is generated by the invertible elements

{1, g1, g2, …, gn-1}

which satisfy the following relations:

gigj = gjgi  if |i-j| > 1,

gigi+1gi= gi+1gigi+1, for i=1, 2, …, n-2,

ei gi = r-1ei, for i=1, 2, …, n-1,

eigi-1ei = rei,

ei(gi-1)-1ei = r-1ei,

where each ei is defined by

(q-q-1)(1-ei) = gi – (gi)-1

Now the really great thing about the Birman-Wenzl-Murakami algebra is that you can obtain knot invariants from it, following Hans Wenzl’s paper
Quantum groups and subfactors of type B, C and D, which can be found in all good bookstores and on Mathscinet.

A key point is that if you have a representation of the Birman-Wenzl-Murakami algebra, you automatically have a knot (link) invariant.

It is well-known that the BWM algebra (for short) is related to (subalgebras of) the centraliser algebra of tensor products of the fundamental represenatations of the quantum algebras Uqso(2r+1) and also Uqsp(2r).

In my PhD thesis I showed that it was also related to the centraliser algebra of tensor products of the fundamental representation of the quantum superalgebra Uqosp(1|2r).

I’ll write more about in this in a futher post.

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