Determining the doubly slice genera of prime knots with up to 12 crossings

TL;DR

This paper computes the doubly slice genera for most prime knots with up to 12 crossings, providing a comprehensive classification of these invariants for a large knot set.

Contribution

The paper determines the doubly slice genera for 2909 prime knots with up to 12 crossings, expanding the known data on this knot invariant.

Findings

Doubly slice genera are identified for 2909 prime knots with ≤12 crossings.

Most prime knots with up to 12 crossings have known doubly slice genera.

The results provide a substantial dataset for future research on knot slicing properties.

Abstract

For a knot $K$, the doubly slice genus $g_{ds}(K)$ is the minimal $g$ such that $K$ divides a closed, orientable, and unknotted surface of genus $g$ embedded in $S^4$. In this paper, we identify the doubly slice genera of 2909 of the 2977 prime knots which have a crossing number of 12 or fewer.