Work

Slope norm and an algorithm to compute the crosscap number

Year: 2024

Type: article

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 I 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.57K10, 57K31; 57K32

Source: Algebraic & Geometric Topology

DOI: 10.2140/agt.2024.24.4307

Authors: William Jaco, Joachim Hyam Rubinstein, Jonathan Spreer, Stephan Tillmann

Institutions: Oklahoma State University, The University of Melbourne, École Polytechnique Fédérale de Lausanne, The University of Sydney

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funder: Australian Research Council

Award: Trisections, triangulations and the complexity of manifolds