Gordian Adjacency for Positive Braid Knots

TL;DR

This paper investigates Gordian adjacency among positive braid knots, providing conditions for adjacency, analyzing unknotting sequences, and proving finiteness results related to unknotting numbers.

Contribution

It extends previous work by establishing new sufficient conditions for Gordian adjacency in positive braid knots and proves finiteness of positive braid knots for fixed unknotting number.

Findings

Sufficient conditions for Gordian adjacency in positive braid knots

Finiteness of positive braid knots with a given unknotting number

Analysis of unknotting sequences for positive braid knots

Abstract

A knot $K_1$ is said to be Gordian adjacent to a knot $K_2$ if $K_1$ is an intermediate knot on an unknotting sequence of $K_2$. We extend previous results on Gordian adjacency by showing sufficient conditions for Gordian adjacency between classes of positive braid knots through manipulations of braid words. In addition, we explore unknotting sequences of positive braid knots and give a proof that there are only finitely many positive braid knots for a given unknotting number.