On surgery curves for genus one slice knots

TL;DR

This paper investigates the relationship between genus one slice knots and certain essential curves on their Seifert surfaces, providing counterexamples and exploring the interplay of invariants to understand sliceness conditions.

Contribution

It presents a counterexample to a conjecture about algebraic sliceness of curves on Seifert surfaces and analyzes the role of Casson-Gordon invariants in this context.

Findings

Counterexample to the conjecture about algebraically slice curves

Relationship established between Casson-Gordon invariants and algebraic invariants of curves

Answers negatively a question of Cooper and relates to Kauffman's problem

Abstract

If a knot K bounds a genus one Seifert surface F in the 3-sphere and F contains an essential simple closed curve alpha that has induced framing 0 and is smoothly slice, then K is smoothly slice. Conjecturally, the converse holds. It is known that if K is slice, then there are strong constraints on the algebraic concordance class of such alpha, and it was thought that these constraints might imply that alpha is at least algebraically slice. We present a counterexample; in the process we answer negatively a question of Cooper and relate the result to a problem of Kauffman. Results of this paper depend on the interplay between the Casson-Gordon invariants of K and algebraic invariants of alpha.