The stable 4-genus of knots

TL;DR

This paper introduces the concept of stable 4-genus for knots, explores its properties, and demonstrates its potential to take noninteger values, revealing new insights into knot concordance and distinctions between smooth and topological categories.

Contribution

It develops the theory of stable 4-genus, provides examples illustrating its properties, and shows it can be noninteger, unlike classical invariants.

Findings

Stable 4-genus induces a seminorm on the concordance group tensor with rationals.

Examples show the stable genus can distinguish smooth and topological categories.

Casson-Gordon invariants reveal the stable genus can be noninteger.

Abstract

The stable 4-genus of a knot K in 3-space is the limiting value of g_4(nK)/n, where g_4 denotes the 4-genus and n goes to infinity. This induces a seminorm on CQ, the concordance group tensored with the rational numbers. Basic properties of the stable genus are developed, as are examples focused on understanding the unit ball under this seminorm for specified subspaces of CQ. Subspaces spanned by torus knots are used to illustrate the distinction between the smooth and topological categories. A final example is given in which Casson-Gordon invariants are used to demonstrate that the stable genus can be a noninteger, something that cannot be detected by classical invariants or those arising from Heegaard-Floer or Khovanov homology. It is unknown if the stable genus determines a norm on CQ and no noninteger value for the stable genus is known.