Nonorientable surfaces bounded by knots: a geography problem

TL;DR

This paper investigates the possible pairs of Euler class and Betti number for nonorientable surfaces bounded by knots, focusing on torus knots and employing Heegaard Floer invariants to refine bounds on the nonorientable 4-genus.

Contribution

It formulates a geography problem for nonorientable surfaces bounded by knots and applies Heegaard Floer invariants to improve bounds for specific knot families.

Findings

Established relationships between Betti number and Euler class for nonorientable surfaces

Identified realizable pairs for torus knots within the geography problem

Used Ozsváth-Szabó invariants to refine bounds on nonorientable 4-genus

Abstract

The nonorientable 4-genus is an invariant of knots which has been studied by many authors, including Gilmer and Livingston, Batson, and Ozsv\'{a}th, Stipsicz, and Szab\'{o}. Given a nonorientable surface $F \subset B^4$ with $\partial F = K\subset S^3$ a knot, an analysis of the existing methods for bounding and computing the nonorientable 4-genus reveals relationships between the first Betti number $\beta_1$ of $F$ and the normal Euler class $e$ of $F$. This relationship yields a geography problem: given a knot $K$, what is the set of realizable pairs $(e(F), \beta_1(F))$ where $F\subset B^4$ is a nonorientable surface bounded by $K$? We explore this problem for families of torus knots. In addition, we use the Ozsv\'ath-Szab\'o $d$-invariant of two-fold branched covers to give finer information on the geography problem. We present an infinite family of knots where this information provides an improvement upon the bound given by Ozsv\'ath, Stipsicz, and Szab\'o using the Upsilon invariant.