On classical upper bounds for slice genera

TL;DR

This paper introduces the algebraic genus, a new link invariant that provides an upper bound for the topological slice genus and relates to various classical invariants, offering insights into knot and link topology.

Contribution

The paper defines the algebraic genus and characterizes it via cobordisms, connecting it to existing invariants and proposing it as an optimal upper bound for the topological slice genus.

Findings

Algebraic genus bounds the topological slice genus.

Connections established between algebraic genus and classical invariants.

Casson-Gordon invariants support algebraic genus as an optimal bound.

Abstract

We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebraic genus as a candidate for the optimal upper bound for the topological slice genus that is determined by the S-equivalence class of Seifert matrices.