Optimal Cobordisms between Torus Knots

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.