The Concordance Genus of 11--Crossing Knots

TL;DR

This paper determines the concordance genus for most prime 11-crossing knots, providing new insights into their concordance classes and including diagrams for knots concordant to lower genus knots.

Contribution

It resolves the concordance genus for 533 out of 552 prime 11-crossing knots, offering detailed diagrams and null-concordance results.

Findings

Concordance genus is bounded by knot and slice genus.

533 knots' concordance genus is determined.

Diagrams for knots concordant to lower genus are provided.

Abstract

The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11--crossing prime knots. This analysis resolves the concordance genus of 533 of the 552 prime 11--crossing knots. The appendix to the paper gives concordance diagrams for 59 knots found to be concordant to knots of lower genus, including null-concordances for the 30 11--crossing knots known to be slice.