Slope norm and an algorithm to compute the crosscap number

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 H1(M ; Z2), determines the maximal Euler characteristic of any properly embedded surface having a boundary curve of this slope. We complement our theoretical wo..