On the algebraic Gordian distance

TL;DR

This paper explores the algebraic Gordian distance between knots using Blanchfield pairings, providing criteria based on quadratic equations for when certain Alexander polynomials can be realized with minimal crossing changes.

Contribution

It introduces a novel algebraic criterion involving quadratic equations and Blanchfield pairings to analyze Gordian distances between knots.

Findings

Certain Alexander polynomials cannot be realized with Gordian distance one without integer solutions.

The results assist in calculating algebraic Gordian distances and polynomial distances.

An example demonstrates the practical application of the theoretical results.

Abstract

Using Blanchfield pairings, we show that two Alexander polynomials cannot be realized by a pair of matrices with Gordian distance one if a corresponding quadratic equation does not have an integer solution. We also give an example of how our results help in calculating the Gordian distances, algebraic Gordian distances and polynomial distances.