Blanchfield forms and Gordian distance

TL;DR

This paper develops new lower bounds on the Gordian distance and unlinking number of links using Alexander module invariants and the Blanchfield pairing, generalizing previous results and answering a longstanding question.

Contribution

It introduces invariants from the Alexander module and Blanchfield pairing to bound link distances, extending Kawauchi's results and addressing a question by Colin Adams.

Findings

Derived new lower bounds on Gordian distance and unlinking number.

Provided restrictions on knot types from splitting operations.

Answered a question posed by Colin Adams in 1996.

Abstract

Given a link in $S^3$ we will use invariants derived from the Alexander module and the Blanchfield pairing to obtain lower bounds on the Gordian distance between links, the unlinking number and various splitting numbers. These lower bounds generalise results recently obtained by Kawauchi. We give an application restricting the knot types which can arise from a sequence of splitting operations on a link. This allows us to answer a question asked by Colin Adams in 1996.