Gordian distance and Vassiliev invariants

TL;DR

This paper demonstrates that for any two knots with Gordian distance two, one can find knots with any specified Vassiliev invariants between them, extending known results about knot transformations.

Contribution

It generalizes the understanding of knot invariants by showing the existence of knots with arbitrary Vassiliev invariants between pairs of knots at Gordian distance two.

Findings

Existence of knots with prescribed Vassiliev invariants between distance-two knot pairs

Extension of previous results on infinite knots between such pairs

Deeper insight into the structure of knot invariants and transformations

Abstract

The Gordian distance between two knots measures how many crossing changes are needed to transform one knot into the other. It is known that there are always infinitely many non-equivalent knots `between' a pair of knots of Gordian distance two. In this paper we prove an extreme generalisation of this fact: there are knots with arbitrarily prescribed Vassiliev invariants between every pair of knots of Gordian distance two.