The non-orientable 4-genus for knots with 8 or 9 crossings

TL;DR

This paper computes the non-orientable 4-genus for all knots with 8 or 9 crossings, confirming a conjecture and providing new bounds for the slicing number, advancing understanding of knot surfaces in four-dimensional space.

Contribution

It provides the first complete calculation of the non-orientable 4-genus for knots with 8 or 9 crossings, confirming a conjecture and establishing a new lower bound for the slicing number.

Findings

Confirmed Murakami's and Yasuhara's conjecture.

Computed non-orientable 4-genus for all 8- and 9-crossing knots.

Established a new lower bound for the slicing number.

Abstract

The non-orientable 4-genus of a knot in the 3-sphere is defined as the smallest first Betti number of any non-orientable surface smoothly and properly embedded in the 4-ball, with boundary the given knot. We compute the non-orientable 4-genus for all knots with crossing number 8 or 9. As applications we prove a conjecture of Murakami's and Yasuhara's, and give a new lower bound for the slicing number of knot.