The nonorientable four-genus of knots

TL;DR

This paper introduces new obstructions, using advanced topological invariants, to determine when knots in the 3-sphere can bound nonorientable surfaces like Klein bottles in the 4-ball, revealing limitations on such embeddings.

Contribution

It develops novel obstructions based on Heegaard-Floer homology and classical theorems, extending understanding of nonorientable surface bounds for knots in four-dimensional topology.

Findings

Obstructions based on linking forms of branched covers.

Stronger obstructions using Ozsvath-Szabo correction terms.

Existence of knots not bounding low Betti number nonorientable surfaces.

Abstract

We develop obstructions to a knot K in the 3-sphere bounding a smooth punctured Klein bottle in the 4-ball. The simplest of these is based on the linking form of the 2-fold branched cover of the 3-sphere branched over K. Stronger obstructions are based on the Ozsvath-Szabo correction term in Heegaard-Floer homology, along with the G-signature theorem and the Guillou-Marin generalization of Rokhlin's theorem. We also apply Casson-Gordon theory to show that for every n greater than one there exists a knot that does not bound a topologically embedded nonorientable ribbon surface F in the 4-ball with first Betti number less than n.