Optimal Cobordisms between Torus Knots

TL;DR

This paper constructs small genus cobordisms between torus knots using algebraic curves and employs the Upsilon-invariant to determine cobordism distances, advancing understanding of knot cobordisms and singularity realizations.

Contribution

It introduces explicit algebraic-curve-based cobordisms between torus knots and computes the Upsilon-invariant for braid index 3 and 4 knots, providing new tools for cobordism analysis.

Findings

Constructed small genus cobordisms between specific torus knots.

Computed the Upsilon-invariant for braid index 3 and 4 torus knots.

Determined cobordism distances for knots of small braid index.

Abstract

We construct cobordisms of small genus between torus knots and use them to determine the cobordism distance between torus knots of small braid index. In fact, the cobordisms we construct arise as the intersection of a smooth algebraic curve in $\mathbb{C}^2$ with the unit 4-ball from which a 4-ball of smaller radius is removed. Connections to the realization problem of $A_n$-singularities on algebraic plane curves and the adjacency problem for plane curve singularities are discussed. To obstruct the existence of cobordisms, we use Ozsv\'ath, Stipsicz, and Szab\'o's $\Upsilon$-invariant, which we provide explicitly for torus knots of braid index 3 and 4.