Counterexamples to Kauffman's Conjectures on Slice Knots

TL;DR

This paper presents counterexamples to Kauffman's conjecture by constructing slice knots with Seifert surfaces where all essential curves have non-zero Arf invariants and signatures, challenging previous assumptions.

Contribution

It provides the first known counterexamples to Kauffman's conjecture, showing that slice knots can have Seifert surfaces with all essential curves non-zero in invariants.

Findings

Counterexamples to Kauffman's conjecture are constructed.

Slice knots can have Seifert surfaces with all essential curves having non-zero invariants.

The results challenge previous support for the conjecture.

Abstract

In 1982 Louis Kauffman conjectured that if a knot in the 3-sphere is a slice knot then on any Seifert surface for that knot there exists a homologically essential simple closed curve of self-linking zero which is itself a slice knot, or at least has Arf invariant zero. Since that time, considerable evidence has been amassed in support of this conjecture. In particular, many invariants that obstruct a knot from being a slice knot have been explictly expressed in terms of invariants of such curves on the Seifert surface. We give counterexamples to Kauffman's conjecture, that is, we exhibit (smoothly) slice knots that admit (unique minimal genus) Seifert surfaces on which every homologically essential simple closed curve of self-linking zero has non-zero Arf invariant and non-zero signatures.