Cosmetic crossings of genus one knots

TL;DR

This paper investigates conditions under which genus one knots admit cosmetic crossings, using Alexander polynomial and double cover homology, and proves the nugatory crossing conjecture for specific knot families.

Contribution

It introduces new obstructions for cosmetic crossings in genus one knots and verifies the nugatory crossing conjecture for several classes of knots.

Findings

Alexander polynomial and double cover homology obstruct cosmetic crossings

Proved the nugatory crossing conjecture for certain Whitehead doubles

Verified the conjecture for some pretzel knots and all genus one knots up to 10 crossings

Abstract

We show that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for the negatively twisted, positive Whitehead doubles of all knots. We also verify the conjecture for several families of pretzel knots and all genus one knots with up to 10 crossings.