Singular Hecke algebras, Markov traces, and HOMFLY-type invariants

Abstract

We define the singular Hecke algebra $\mathcal{H}(S{B}_n)$ as the quotient of the singular braid monoid algebra $\mathbb{C}(q)[S{B}_n]$ by the Hecke relations ${\sigma }_k^2=(q-1){\sigma }_k+q$, $1\le k\le n-1$, and define the Markov traces on the sequence $\{\mathcal{H}(S{B}_n){\}}_{n=1}^{+\infty }$ in the same way as for the Markov traces on the tower of (non-singular) Hecke algebras of the symmetric groups. We prove that the Markov traces are in one-to-one correspondance with the invariants that satisfies some skein relation, and compute an explicit classification of the Markov traces. Thanks to this classification, we define some universal HOMFLY-type invariant which has the property that it distinguishes all the pairs of singular links that can be distinguished by an invariant which satisfies the required skein relation.