Everywhere Equivalent 3-Braids

TL;DR

This paper classifies all closed 3-braid diagrams where changing any single crossing results in the same knot or link, revealing a specific class of knots with this property.

Contribution

It provides a complete classification of everywhere equivalent diagrams within the class of closed 3-braids, a novel result in knot theory.

Findings

Identified all closed 3-braid diagrams with the property that crossing changes yield the same knot or link.

Established criteria for when a 3-braid diagram is everywhere equivalent.

Contributed to understanding the symmetry and structure of knots related to crossing modifications.

Abstract

A knot (or link) diagram is said to be everywhere equivalent if all the diagrams obtained by switching one crossing represent the same knot (or link). We classify such diagrams of a closed 3-braid.