Minimal Cobordisms between Torus Knots

Peter Feller

Published 2015-01-02Version 1

We determine the cobordism distance between torus knots of small braid index. We also provide new minimal cobordisms between the torus knots $T_{m, m+1}$ and torus knots of braid index two. In fact, all minimal 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 simple 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.