Ribbon concordances and slice obstructions: experiments and examples

TL;DR

This paper investigates the sliceness of prime knots with up to 19 crossings, providing extensive computational results and new methods to determine smooth and topological sliceness, revealing that most such knots are not slice.

Contribution

The authors develop and apply new techniques to determine sliceness for a vast number of knots, producing a large dataset and insights into conjectures in knot theory.

Findings

Approximately 1.6 million knots are smoothly slice (ribbon)

About 350.5 million knots are not topologically slice

The study introduces new methods for probing sliceness properties

Abstract

There are 352.2 million prime knots in the 3-sphere with at most 19 crossings. We study which of these knots are slice, in both the smooth and topological categories. While no algorithm is known for deciding whether a given knot is slice in either setting, we are able to determine it smoothly for all but about 11,400 knots (0.003% or 1 in 30,000) and topologically for all but about 1,400 knots (0.0004% or about 1 in 250,000). In particular, we show that some 1.6 million of these knots (0.46%) are smoothly slice (in fact ribbon) and that 350.5 million are not even topologically slice (99.54%). We use a wide range of tools and techniques, and introduce several new or refined methods for probing these properties. Along the way, we produce 500,000 pairs of 0-friends, that is, pairs of distinct knots with the same 0-surgery. We discuss how our data is consistent with several important conjectures and suggests new ones, and highlight the simplest knots where sliceness remains unknown.