Cusp areas of Farey manifolds and applications to knot theory

Abstract

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.

44 pages, 11 figures. Version 4 contains revisions and corrections (most notably, in Sections 5 and 6) that incorporate referee comments. To appear in the International Mathematics Research Notices.

References in corpus (5)

• The Jones polynomial and graphs on surfaces
• Alternating sum formulae for the determinant and other link invariants
• Symmetric links and Conway sums: volume and Jones polynomial
• Non-triviality of the Jones polynomial and the crossing numbers of amphicheiral knots
• On links with cyclotomic Jones polynomials

Cited by in corpus (13)

• Symmetric links and Conway sums: volume and Jones polynomial
• Volume bounds for generalized twisted torus links
• Cusp geometry of fibered 3-manifolds
• Cusp volumes of alternating knots
• Jones polynomials, volume, and essential knot surfaces: a survey
• Geometric triangulations of a family of hyperbolic 3-braids
• Monodromy action on unknotting tunnels in fiber surfaces
• Cusp Volumes of Alternating Knots on Surfaces
• Linear Bounds of the Crosscap Number of Knots
• On 3-braids and L-space knots
• Cusp shape and tunnel number
• Waist size for cusps in hyperbolic 3-manifolds II
• Combinatorics of Link Diagrams and Volume