Volumes of highly twisted knots and links

Abstract

We show that for a large class of knots and links with complements in S 3 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. 57M25, 57M501 IntroductionGiven a diagram of a knot or link, our goal is to determine geometric information about the complement of that link in S 3 . In particular, if the complement admits a hyperbolic structure, then by Mostow-Prasad rigidity that structure is unique. We ought to be able to make explicit statements about the geometry of this link complement, including statements about its volume. However, such results based purely on a diagram seem to be rare. For particular classes of knots and links, other results on volume have been determined. Lackenby proved that in the special case in which a knot or link is alternating, the volume of the complement is bounded above and below by the twist number of a diagram [10]. In fact, the upper bound is valid for all knots, not just alternating. This upper bound was further improved by Agol and D Thurston in an appendix to Lackenby's paper. Additionally, they found a sequence of links with volume approaching the upper bound.Recently, the lower bound has been improved by work of Agol, Storm and Thurston [2]. The proof of this result still requires that the links in question be alternating, however.

Dehn filling, volume, and the Jones polynomial

“…The hyperbolicity of S 3 L is a consequence of work of Adams [3]. See also Purcell [36]. To estimate the volume of S 3 L, we simplify the link L even further, by removing all remaining single crossings from the twist regions of L. The resulting flat augmented link L ′ has the same volume as L, by the work of Adams [2].…”

Mutations and short geodesics in hyperbolic 3-manifolds

“…A full twist consists of two half twists. Now, we can define augmented links, which were introduced by Adams [1] and have been studied extensively by Futer and Purcell in [11] and Purcell in [33], [34]. For an introduction to augmented links, we suggest first reading [35].…”

Generalized augmented alternating links and hyperbolic volumes

“…Augmented alternating links are useful in obtaining upper bounds on volumes of links. See for instance [6] and [8]. In some papers, the links considered are maximally augmented.…”