On Calculating the Slice Genera of 11- and 12-Crossing Knots

ABSTRACT

This article contains the results of efforts to determine the 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 the 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.

Keywords:

• Knots
• Smooth Slice Genus
• Topological Slice Genus
• Seifert Form
• Donaldson's Diagonalization Theorem

2010 AMS SUBJECT CLASSIFICATION:

• 57M25
• 57M27

Acknowledgments

Duncan McCoy would like to thank his supervisor, Brendan Owens, for his continued guidance. Both authors would like to thank Peter Feller for the conversations that led to the undertaking of this project.

Notes

2 12a412 and 12n595 are both Gordian distance one from a knot with the same Jones polynomial as 11n50, 11n132, and 12n414, which are all slice, and 11n133, which is not. It can be verified that the knot obtained in both cases is, in fact 11n50, confirming that the smooth slice genus is one.