The next simplest hyperbolic knots

  title={The next simplest hyperbolic knots},
  author={Abhijit Champanerkar and Ilya Kofman and Eric D. Patterson},
  journal={arXiv: Geometric Topology},
We complete the project begun by Callahan, Dean and Weeks to identify all knots whose complements are in the SnapPea census of hyperbolic manifolds with seven or fewer tetrahedra. Many of these ``simple'' hyperbolic knots have high crossing number. We also compute their Jones polynomials. 
The 500 simplest hyperbolic knots
We identify all hyperbolic knots whose complements are in the census of orientable one-cusped hyperbolic manifolds with eight ideal tetrahedra. We also compute their Jones polynomials.
Hyperbolic geometry of multiply twisted knots
We investigate the geometry of hyperbolic knots and links whose diagrams have a high amount of twisting of multiple strands. We find information on volume and certain isotopy classes of geodesics for
Dehn filling, volume, and the Jones polynomial
Given a hyperbolic 3–manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammatic
A knot without a nonorientable essential spanning surface
This note gives the first example of a hyperbolic knot in the 3-sphere that lacks a nonorientable essential spanning surface; this disproves the Strong Neuwirth Conjecture formulated by Ozawa and
A-Polynomials of fillings of the Whitehead sister
Knots obtained by Dehn filling the Whitehead sister link include some of the smallest volume twisted torus knots. Here, using results on A-polynomials of Dehn fillings, we give formulas to compute
Unknown values in the table of knots
This paper, to be regularly updated, lists those prime knots with the fewest possible number of crossings for which values of basic knot invariants, such as the unknotting number or the smooth
This paper is a brief overview of recent results by the authors relating colored Jones polynomials to geometric topology. The proofs of these results appear in the papers (18, 19), while this survey
Guts of Surfaces and the Colored Jones Polynomial
Decomposition into 3-balls, Ideal Polyhedra, and I-bundles and essential product disks are presented.
Cusp Areas of Farey Manifolds and Applications to Knot Theory
This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured-torus bundles, 4-punctured sphere bundles, and two-bridge link complements. The input for these estimates
Hyperbolic Knot Theory
This book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. It has three parts. The first part covers


While the crossing number is the standard notion of complexity for knots, the number of ideal tetrahedra required to construct the complement provides a natural alternative. We determine which
A Computer Generated Census of Cusped Hyperbolic 3-Manifolds
This paper describes how a computer was used to produce a census of cusped hyperbolic 3-manifolds obtained from 5 or fewer ideal tetrahedra and gives a brief summary of the results.
Knots and Links
Introduction Codimension one and other matters The fundamental group Three-dimensional PL geometry Seifert surfaces Finite cyclic coverings and the torsion invariants Infinite cyclic coverings and
The Volume of Hyperbolic Alternating Link Complements
If a hyperbolic link has a prime alternating diagram D, then we show that the link complement's volume can be estimated directly from D. We define a very elementary invariant of the diagram D, its
The colored Jones polynomials and the simplicial volume of a knot
We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect non-trivially. Moreover it is shown that the
Word hyperbolic Dehn surgery
In the late 1970’s, Thurston dramatically changed the nature of 3-manifold theory with the introduction of his Geometrisation Conjecture, and by proving it in the case of Haken 3-manifolds [23]. The
4-manifolds and Kirby calculus
4-manifolds: Introduction Surfaces in 4-manifolds Complex surfaces Kirby calculus: Handelbodies and Kirby diagrams Kirby calculus More examples Applications: Branched covers and resolutions Elliptic
Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant
Preface Introduction Glossary 1 Heegaard splittings 1.1 Introduction 1.2 Existence of Heegaard splittings 1.3 Stable equivalence of Heegaard splittings 1.4 The mapping class group 1.5 Manifolds of
Computers and Mathematics
This volume contains the contributed papers accepted for presentation, selected from 85 drafts submitted in response to the call for papers.
4-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics
  • American Mathematical Society,
  • 1999