Slope norm and an algorithm to compute the crosscap number

TL;DR

This paper introduces three algorithms based on normal surface theory to compute the crosscap number of knots, extending to a broader class of 3-manifolds, and successfully computes 196 previously unknown crosscap numbers.

Contribution

The paper presents new algorithms for calculating the crosscap number of knots using 0-efficient triangulations, applicable to a wider class of 3-manifold complements, with practical implementation and new knot data.

Findings

Successfully computed 196 new crosscap numbers

Algorithms are correct for a larger class of knot complements

Implementation demonstrates practical applicability

Abstract

We give three algorithms to determine the crosscap number of a knot in the 3-sphere using $0$-efficient triangulations and normal surface theory. Our algorithms are shown to be correct for a larger class of complements of knots in closed 3-manifolds. The crosscap number is closely related to the minimum over all spanning slopes of a more general invariant, the slope norm. For any irreducible 3-manifold $M$ with incompressible boundary a torus, we give an algorithm that, for every slope on the boundary that represents the trivial class in $H_1(M; \mathbb{Z}_2)$, determines the maximal Euler characteristic of any properly embedded surface having a boundary curve of this slope. We complement our theoretical work with an implementation of our algorithms, and compute the crosscap number of knots for which previous methods would have been inconclusive. In particular, we determine 196 previously unknown crosscap numbers in the census of all knots with up to 12 crossings.