Neighbors of knots in the Gordian graph

TL;DR

This paper demonstrates that any knot can be transformed into another with arbitrarily high bridge number and distance through a single crossing change, revealing new insights into the structure of the Gordian graph.

Contribution

It establishes that every knot is one crossing change away from knots with arbitrarily high bridge number and distance, advancing understanding of knot transformations.

Findings

Any knot is one crossing change away from knots with high bridge number.

Any knot is one crossing change away from knots with high bridge distance.

The result connects local crossing changes to global knot complexity.

Abstract

We show that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrarily high bridge distance.