Exchange moves and Fiedler polynomial

TL;DR

This paper explores how the Fiedler polynomial and related invariants, including a Kauffman bracket in the solid torus, can distinguish conjugacy classes of exchange related braids, advancing knot theory understanding.

Contribution

It introduces finite type invariants derived from the Kauffman bracket in the solid torus and analyzes their effectiveness in distinguishing exchange related braids.

Findings

Fiedler polynomial can distinguish certain conjugacy classes

Kauffman bracket in the solid torus yields finite type invariants

Invariants help differentiate exchange related braids

Abstract

In order to obtain a Markov theorem without stabilization, Birman and Menasco introduced the notion of exchange related braids. In this paper I study the way the Fiedler polynomial distinguishes conjugacy classes of some particular braided knots. I introduce the Kauffman bracket in the solid torus. Its Taylor expansion give finite type invariants similar to the Fiedler polynomial. I investigate how these invariants distinguish exchange related braids.