Constructing Markov traces simply - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T20:59:03Z http://mathoverflow.net/feeds/question/91925 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91925/constructing-markov-traces-simply Constructing Markov traces simply Thirsty 2012-03-22T15:40:26Z 2012-03-22T15:40:26Z <p>Short version: I wondering how to simply check if a proposed Markov trace, $\phi$ had the correct property using techniques similar to those from the Akutsu-Wadati 1987 paper `Exactly solvable models and new link polynomials. I: N-state vertex models'. The property $\phi(AB)=\phi(BA)$ appears to be simple while I am unaware of a simple and general way to check the other property.</p> <p>Long version: My question is about constructing a Markov trace associated with a particular solution to the braid equation. For the braid group $B_{n}$ (for $n$ strands with generators $b_{i}$), the Markov trace, $\phi$, is required to satisfy two properties:</p> <p>1) $\phi(AB) = \phi(BA)$ for $A,B \in B_{n}$,</p> <p>2) $\phi(Ab_{n}) = \tau\phi(A)$ for $A \in B_{n}$ for some $\tau$ independent of $n$.</p> <p>Using the Akutsu-Wadati 1987 paper, the say that given a representation of the generators, $b \rightarrow \tilde{b} \in \mbox{End}(V\otimes V)$ for some vector space $V$, then it is possible to define Markov trace as $\phi(A) = \mbox{Tr}(H\tilde{A})$ ($\tilde{A}$ denotes the representation of the braid word $A$). Tr is the usual matrix trace while $H=h\otimes h \cdots \otimes h$ with $h\in \mbox{End}(V)$. To test the first property one needs merely to check $$\left[\tilde{b}, h\otimes h\right] = 0.$$</p> <p>Does anyone know of a simple test for the second condition? I think $$\mbox{id} \otimes \mbox{Tr} [\tilde{b} (I_{V} \otimes h)] \propto I_{V},$$ is sufficient but from what I have read this is not true for the Jones polynomial.</p> <p>If anyone knows of other simple systematic methods for constructing link invariants I would be interested to find out. Also, would the procedure generalise if the representation mapped $b \rightarrow \tilde{b} \in \mbox{End}(V\otimes V\otimes V)$</p>