Fundamental normal surfaces and the enumeration of Hilbert bases

Details Fundamental normal surfaces and the enumeration of Hilbert bases Fundamental normal surfaces and the enumeration of Hilbert bases

2011-11-30

arxiv.org/abs/1111.7055v1

Mathematics

Geometric Topology

18 pages, 3 figures

Scientific paper

Normal surfaces are a key tool in computational knot theory and 3-manifold topology, and have featured in significant computational breakthroughs in recent years. Despite this, there has been little practical progress on algorithms that use fundamental normal surfaces, which are described in terms of a Hilbert basis for a pointed rational cone on a high-dimensional integer lattice. In this paper we develop and implement several algorithms to enumerate fundamental normal surfaces, by merging domain-specific techniques from normal surface theory with classical Hilbert basis algorithms. The most successful of these combines a maximal admissible face decomposition with the primal Hilbert basis algorithm of Bruns, Ichim and Koch, and in many cases can solve 168-dimensional enumeration problems (based on 24-tetrahedron knot complements) in a matter of hours. As an application, we use this new algorithm to compute 164 previously unknown crosscap numbers in the KnotInfo database of knot invariants.

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