On the nonorientable 4-genus of double twist knots

Abstract

We investigate the nonorientable 4-genus [Formula: see text] of a special family of 2-bridge knots, the double twist knots [Formula: see text]. Because the nonorientable 4-genus is bounded by the nonorientable 3-genus, it is known that [Formula: see text]. By using explicit constructions to obtain upper bounds on [Formula: see text] and known obstructions derived from Donaldson’s diagonalization theorem to obtain lower bounds on [Formula: see text], we produce infinite subfamilies of [Formula: see text] where [Formula: see text] and [Formula: see text], respectively. However, there remain infinitely many double twist knots where our work only shows that [Formula: see text] lies in one of the sets [Formula: see text], or [Formula: see text]. We tabulate our results for all [Formula: see text] with [Formula: see text] and [Formula: see text] up to 50. We also provide an infinite number of examples which answer a conjecture of Murakami and Yasuhara.