Slope norm and an algorithm to compute the crosscap number

William Jaco, J. Hyam Rubinstein, Jonathan Spreer, Stephan Tillmann

Published 2021-08-17Version 1

We give an algorithm to determine the crosscap number of a knot in the 3-sphere using $0$-efficient triangulations and normal surface theory. This algorithm is 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 204 previously unknown crosscap numbers in the census of all knots with up to 12 crossings.