Geodesic
Slope norm and an algorithm to compute the crosscap number
Jaco, William1 ; Hyam Rubinstein, Joachim2 ; Spreer, Jonathan3 ; Tillmann, Stephan3
1 Department of Mathematics, Oklahoma State University, Stillwater, OK, United States
2 School of Mathematics and Statistics, The University of Melbourne, Melbourne VIC, Australia
3 School of Mathematics and Statistics, The University of Sydney, Sydney NSW, Australia
Algebraic and Geometric Topology, Tome 24 (2024) no. 8, pp. 4307-4351

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

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 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.

DOI : 10.2140/agt.2024.24.4307
Keywords: $3$–manifold, knot, efficient triangulation, crosscap number, knot genus, boundary slope, slope norm

Jaco, William 1 ; Hyam Rubinstein, Joachim 2 ; Spreer, Jonathan 3 ; Tillmann, Stephan 3

1 Department of Mathematics, Oklahoma State University, Stillwater, OK, United States
2 School of Mathematics and Statistics, The University of Melbourne, Melbourne VIC, Australia
3 School of Mathematics and Statistics, The University of Sydney, Sydney NSW, Australia 
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Jaco, William; Hyam Rubinstein, Joachim; Spreer, Jonathan; Tillmann, Stephan. Slope norm and an algorithm to compute the crosscap number. Algebraic and Geometric Topology, Tome 24 (2024) no. 8, pp. 4307-4351. doi: 10.2140/agt.2024.24.4307

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