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

TL;DR

This paper introduces a new algebraic framework for singular links using singular Hecke algebras, classifies Markov traces, and constructs a universal HOMFLY-type invariant capable of distinguishing all singular links detectable by such invariants.

Contribution

It defines singular Hecke algebras, classifies all associated Markov traces, and constructs a universal HOMFLY-type invariant for singular links.

Findings

Classified all Markov traces on singular Hecke algebras.

Established a correspondence between Markov traces and skein relation invariants.

Constructed a universal HOMFLY-type invariant for singular links.

Abstract

We define the singular Hecke algebra ${\mathcal H} (SB_n)$ as the quotient of the singular braid monoid algebra ${\mathbb C} (q) [SB_n]$ 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 $\{{\mathcal H}(SB_n)\}_{n=1}^{+\infty}$ 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.