Geodesic
Volumes of highly twisted knots and links
Purcell, Jessica1
1Department of Mathematics, 1 University Station C1200, University of Texas at Austin, Austin, TX 78712
Algebraic and Geometric Topology, Tome 7 (2007) no. 1, pp. 93-108
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We show that for a large class of knots and links with complements in S3 admitting a hyperbolic structure, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a knot or link admits a prime, twist reduced diagram with at least 2 twist regions and at least C crossings per twist region, then the link complement is hyperbolic with volume bounded below by 3.3515 times the number of twist regions in the diagram. C is at most 113.

DOI : 10.2140/agt.2007.7.93
Keywords: hyperbolic knot complements, hyperbolic link complements, volume, cone manifolds

Purcell, Jessica  1

1 Department of Mathematics, 1 University Station C1200, University of Texas at Austin, Austin, TX 78712 
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Purcell, Jessica. Volumes of highly twisted knots and links. Algebraic and Geometric Topology, Tome 7 (2007) no. 1, pp. 93-108. doi: 10.2140/agt.2007.7.93

[1] C C Adams, Augmented alternating link complements are hyperbolic, from: "Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984)", London Math. Soc. Lecture Note Ser. 112, Cambridge Univ. Press (1986) 115

[2] I Agol, N Dunfield, P Storm, W P Thurston, Lower bounds on volumes of hyperbolic Haken 3-manifolds

[3] K Böröczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32 (1978) 243

[4] C Cao, G R Meyerhoff, The orientable cusped hyperbolic $3$-manifolds of minimum volume, Invent. Math. 146 (2001) 451

[5] D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5, Mathematical Society of Japan (2000)

[6] D Futer, J S Purcell, Links with no exceptional surgeries

[7] C Hodgson, Degeneration and regeneration of geometric structures on $3$-manifolds, PhD thesis, Princeton Univ. (1986)

[8] C D Hodgson, S P Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998) 1

[9] C D Hodgson, S P Kerckhoff, Universal bounds for hyperbolic Dehn surgery, Ann. of Math. $(2)$ 162 (2005) 367

[10] M Lackenby, The volume of hyperbolic alternating link complements, Proc. London Math. Soc. $(3)$ 88 (2004) 204

[11] C J Leininger, Small curvature surfaces in hyperbolic 3-manifolds, J. Knot Theory Ramifications 15 (2006) 379

[12] J Purcell, Cusp shapes under cone deformation

[13] J Purcell, Cusp Shapes of Hyperbolic Link Complements and Dehn Filling, PhD thesis, Stanford University (2004)

[14] D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish (1976)

[15] W Thurston, The Geometry and Topology of Three-Manifolds, Princeton Univ. Math. Dept. Notes (1979)

[16] J Weeks, Snappea

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