Torsion numbers of augmented groups: with applications to knots and links

2002/02/19 by Daniel S. Silver, Susan G. Williams, Silver, Daniel S. +1 · 1 citation
Mathematics · #11R06 #37B10 #57Q45 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:11R06 #msc:37B10 #msc:57Q45

paper · pdf · doi:10.48550/arxiv.math/0202197

24 pages, no figures. Small corrections and amplifications of previous version

arxiv created 2002/08/06 · arxiv updated 2009/11/30

Abstract

Torsion and Betti numbers for knots are special cases of more general invariants associated to a finitely generated group G and epimorphism from G to the integers. The sequence of Betti numbers is always periodic; under mild hypotheses, the sequence of torsion numbers satisfies a linear homogeneous recurrence relation with constant coeffiencts. Generally, the torsion number sequence exhibits exponential growth rate. However, again under mild hypotheses, the p-part has trivial growth for any prime p. Applications to branched cover homology for knots and links are presented.

Cited by

• Homological torsion growth in non-normal chains of graphs of free groups

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• Betti numbers and torsions in homology groups of double coverings
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• Generalized torsion for knots with arbitrarily high genus