Cusp areas of Farey manifolds and applications to knot theory

Details Cusp areas of Farey manifolds and applications to knot theory Cusp areas of Farey manifolds and applications to knot theory

2008-08-20

arxiv.org/abs/0808.2716v4

Int Math Res Notices 2010, Issue 23, 4434-4497

Mathematics

Geometric Topology

44 pages, 11 figures. Version 4 contains revisions and corrections (most notably, in Sections 5 and 6) that incorporate refere

Scientific paper

10.1093/imrn/rnq037

This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to explicit bounds on the volume of Dehn fillings of these manifolds, for example sharp bounds on volumes of hyperbolic closed 3-braids in terms of the Schreier normal form of the associated braid word. Finally, these results are applied to derive relations between the Jones polynomial and the volume of hyperbolic knots, and to disprove a related conjecture.

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