An invariant for singular knots

TL;DR

This paper introduces a new Jones-type invariant for singular knots using Yokonuma--Hecke algebras and singular braids, expanding knot invariants to include singular crossings with a specific trace condition.

Contribution

It develops a novel invariant for singular knots based on Yokonuma--Hecke algebras and establishes a skein relation involving singular crossings.

Findings

The invariant is constructed via a Markov trace on Yokonuma--Hecke algebras.

A specific trace condition is necessary for the invariant to be well-defined.

The invariant satisfies a skein relation derived from algebraic relations.

Abstract

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma--Hecke algebras ${\rm Y}_{d,n}(u)$ and the theory of singular braids. The Yokonuma--Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid $SB_n$ into the algebra ${\rm Y}_{d,n}(u)$. Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra ${\rm Y}_{d,n}(u)$.