The Conway knot is not slice

TL;DR

This paper proves that the Conway knot is not slice, resolving a long-standing question and completing the classification of slice knots with fewer than 13 crossings, while also providing a novel example of a topologically slice mutant knot.

Contribution

It establishes that the Conway knot is not slice, marking the first such example of a topologically slice mutant of a slice knot, and completes the classification of small crossing slice knots.

Findings

Conway knot is not slice.

First example of a topologically slice mutant of a slice knot.

Completes classification of slice knots under 13 crossings.

Abstract

A knot is said to be slice if it bounds a smooth properly embedded disk in the 4-ball. We demonstrate that the Conway knot, 11n34 in the Rolfsen tables, is not slice. This completes the classification of slice knots under 13 crossings, and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.