Cubic Hecke Algebras and Invariants of Transversal Links

TL;DR

This paper introduces an algebraic method using cubic Hecke algebras to create invariants for transversal links, extending Jones' approach for classical links, and provides an algorithm for practical computation.

Contribution

It develops a new algebraic framework with a semi-Markov trace on cubic Hecke algebras for transversal link invariants, generalizing Jones' method.

Findings

Constructed a semi-Markov trace invariant under positive Markov moves.

Defined an algebraic structure for transversal link invariants.

Provided an algorithm for Groebner basis computation of the ideal.

Abstract

We propose a purely algebraic approach to construct invariants of transversal links in the standard contact structure on the 3-sphere generalizing Jones' approach to invariant of usual links. The only geometry used is the analogue of Alexander and Markov theorems. More precisely, we construct a trace on a certain cubic Hecke algebra which is invariant under positive Markov moves only (we propose to call it a semi-Markov trace). The trace takes its values in the quotient of a polynomial ring by a certain ideal. An algorithm for computing a Groebner base of the ideal is given.