Crossing changes in closed 3-braid diagrams

TL;DR

This paper proves that in closed 3-braid diagrams, nugatory crossings imply a low braid index, supporting a conjecture about nugatory crossings in knot theory.

Contribution

It introduces an invariant for closed 3-braid diagrams and proves a special case of Lin's conjecture relating nugatory crossings to braid index.

Findings

Nugatory crossings in closed 3-braids imply braid index one or two.

The invariant distinguishes nugatory crossings in 3-braid diagrams.

Supports a conjecture linking nugatory crossings to braid index in knot theory.

Abstract

A crossing in a knot is nugatory if changing the crossing does not change the knot type. Using an invariant of certain types of closed 3-braid diagrams, we show that if a closed 3-braid contains a nugatory crossing then its braid index is one or two. This proves a special case of a conjecture on nugatory crossings due to Xiao-Song Lin.