On the nonorientable 4-genus of double twist knots

TL;DR

This paper studies the nonorientable 4-genus of double twist knots, providing bounds, explicit examples, and partial classifications for an infinite family of knots, and confirming a conjecture in the field.

Contribution

It introduces new bounds and explicit constructions for the nonorientable 4-genus of double twist knots, advancing understanding of their topological properties.

Findings

Identified infinite subfamilies with specific nonorientable 4-genus values

Produced explicit constructions for upper bounds on $ ext{4}$-genus

Confirmed a conjecture of Murakami and Yasuhara

Abstract

We investigate the nonorientable 4-genus $\gamma_4$ of a special family of 2-bridge knots, the twist knots and double twist knots $C(m,n)$. Because the nonorientable 4-genus is bounded by the nonorientable 3-genus, it is known that $\gamma_4(C(m,n)) \le 3$. By using explicit constructions to obtain upper bounds on $\gamma_4$ and known obstructions derived from Donaldson's diagonalization theorem to obtain lower bounds on $\gamma_4$, we produce infinite subfamilies of $C(m,n)$ where $\gamma_4=0,1,2,$ and $3$, respectively. However, there remain infinitely many double twist knots where our work only shows that $\gamma_4$ lies in one of the sets $\{1,2\}, \{2,3\}$, or $\{1,2,3\}$. We tabulate our results for all $C(m,n)$ with $|m|$ and $|n|$ up to 50. We also provide an infinite number of examples which answer a conjecture of Murakami and Yasuhara.