Kauffman type invariants for tied links

TL;DR

This paper introduces two new invariants for tied links, extending classical polynomials and demonstrating greater power in distinguishing links, along with a new algebraic framework and connections to Markov traces.

Contribution

It defines novel invariants for tied links extending Kauffman and Jones polynomials, and introduces a new algebraic structure analogous to BMW algebra.

Findings

New invariants outperform classical polynomials on links.

Extension of Kauffman polynomial surpasses Homflypt and recent invariants.

A new algebraic structure is proposed with a recoverable Markov trace.

Abstract

We define two new invariants for tied links. One of them can be thought as an extension of the Kauffman polynomial and the other one as an extension of the Jones polynomial which is constructed via a bracket polynomial for tied links. These invariants are more powerful than both the Kauffman and the bracket polynomials when evaluated on classical links. Further, the extension of the Kauffman polynomial is more powerful of the Homflypt polynomial, as well as of certain new invariants introduced recently. Also we propose a new algebra which plays in the case of tied links the same role as the BMW algebra for the Kauffman polynomial in the classical case. Moreover, we prove that the Markov trace on this new algebra can be recovered from the extension of the Kauffman polynomial defined here.