On calculating the slice genera of 11- and 12-crossing knots

TL;DR

This paper determines the smooth and topological slice genera of most 11- and 12-crossing knots using computational searches, algebraic methods, and new obstructions, significantly advancing the classification of these knot invariants.

Contribution

It provides the first comprehensive computation of topological slice genera for all knots up to 11 crossings and narrows down the unknowns for 12-crossing knots, introducing new obstructions and computational techniques.

Findings

Complete topological slice genus for all knots up to 11 crossings.

Only 2 11-crossing knots remain with unknown smooth genus.

25 12-crossing knots still have unknown slice genus values.

Abstract

This paper contains the results of efforts to determine values of the smooth and the topological slice genus of 11- and 12-crossing knots. Upper bounds for these genera were produced by using a computer to search for genus one concordances between knots. For the topological slice genus further upper bounds were produced using the algebraic genus. Lower bounds were obtained using a new obstruction from the Seifert form and by use of Donaldson's diagonalization theorem. These results complete the computation of the topological slice genera for all knots with at most 11 crossings and leaves the smooth genera unknown for only two 11-crossing knots. For 12 crossings there remain merely 25 knots whose smooth or topological slice genus is unknown.