Gordian Distance and Complete Alexander Neighbors

TL;DR

This paper investigates the properties of complete Alexander neighbors, knots that realize all Alexander polynomials through a single crossing change, and explores their existence and related determinant conditions.

Contribution

It eliminates infinite families of knots with nontrivial Alexander polynomials from being complete Alexander neighbors and refines unknotting number data using determinant conditions.

Findings

No known complete Alexander neighbor has a nontrivial Alexander polynomial.

Determinant conditions can improve unknotting number data for certain knots.

Lickorish's obstruction does not encompass all determinant-based obstructions.

Abstract

We call a knot $K$ a complete Alexander neighbor if every possible Alexander polynomial is realized by a knot one crossing change away from $K$. It is unknown whether there exists a complete Alexander neighbor with nontrivial Alexander polynomial. We eliminate infinite families of knots with nontrivial Alexander polynomial from having this property and discuss possible strategies for unresolved cases. Additionally, we use a condition on determinants of knots one crossing change away from unknotting number one knots to improve KnotInfo's unknotting number data on 11 and 12 crossing knots. Lickorish introduced an obstruction to unknotting number one which proves the same result. However, we show that Lickorish's obstruction does not subsume the obstruction coming from the condition on determinants.