Markov trace on the algebra of braids and ties

TL;DR

This paper introduces a new three-parameter invariant for classical and singular knots derived from a Markov trace on the algebra of braids and ties, extending known knot invariants.

Contribution

It proves the existence of a Markov trace on the algebra of braids and ties and constructs new knot invariants extending the Homflypt polynomial.

Findings

Supports a Markov trace on the algebra of braids and ties

Defines new three-parameter knot invariants

Extends known invariants for classical and singular knots

Abstract

We prove that the so-called t algebra of braids and ties supports a Markov trace. Further, by using this trace in the Jones' recipe, we define invariant polynomials for classical knots and singular knots. Our invariants have three parameters. The invariant of classical knots is an extension of the Homflypt polynomial and the invariant of singular knots is an extension of an invariant of singular knots previously defined by S. Lambropoulou and the second author.