Coloring Link Diagrams And Conway-type Polynomial Of Braids

TL;DR

This paper introduces a new combinatorial 3-variable Laurent polynomial invariant for conjugacy classes in braid groups, satisfying the Conway skein relation and capturing Vassiliev invariants.

Contribution

It provides a simple combinatorial formula for a braid invariant that satisfies the Conway skein relation and encodes Vassiliev invariants.

Findings

Defines a 3-variable Laurent polynomial invariant for braids.

Proves the invariant satisfies the Conway skein relation.

Shows the coefficients are Vassiliev invariants.

Abstract

In this paper we define and present a simple combinatorial formula for a 3-variable Laurent polynomial invariant of conjugacy classes in Artin braid group $B_m$. We show that this Laurent polynomial satisfies the Conway skein relation and its coefficients are Vassiliev invariants of braids.