Genus one cobordisms between torus knots

TL;DR

This paper classifies pairs of torus knots connected by genus one cobordisms, using obstructions from Heegaard Floer invariants and explicit constructions, with applications to crossing changes and Legendrian knots.

Contribution

It provides a complete characterization of genus one cobordisms between torus knots, combining Floer-theoretic obstructions with explicit geometric constructions.

Findings

Identifies all torus knot pairs with genus one cobordisms, except one case.

Determines pairs related by a single crossing change.

Classifies Legendrian torus knots with genus one Lagrangian cobordisms, with one exception.

Abstract

We determine the pairs of torus knots that have a genus one cobordism between them, with one notable exception. This is done by combining obstructions using $\nu^+$ from the Heegaard Floer knot complex and explicit constructions of cobordisms. As an application, we determine the pairs of torus knots related by a single crossing change. Also, we determine the pairs of Thurston-Bennequin number maximizing Legendrian torus knots that have a genus one exact Lagrangian cobordism, with one exception.