Optimal cobordisms between torus knots

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 Ozsváth, Stipsicz, and Szabó’s $\Upsilon$-invariant, which we provide explicitly for torus knots of braid index $3$ and $4$.

Keywords

slice genus, four-genus, torus knots, cobordism distance, positive braids, quasi-positive braids and links, Heegaard–Floer concordance invariants, $\Upsilon$-invariant

2010 Mathematics Subject Classification

14B07, 32S55, 57M25, 57M27

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Published 6 March 2017