The non-orientable 4-genus for knots with 10 crossings

TL;DR

This paper computes the non-orientable 4-genus for all knots with 10 crossings, advancing understanding of knot surfaces in four-dimensional topology.

Contribution

It provides the first complete calculation of the non-orientable 4-genus for knots with 10 crossings, filling a gap in knot theory.

Findings

Calculated non-orientable 4-genus for all 10-crossing knots

Identified patterns in non-orientable surface minimal genus

Extended previous results to more complex knots

Abstract

Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this paper, we calculate the non-orientable 4-genus of knots with crossing number 10.