# The next simplest hyperbolic knots

@article{Champanerkar2003TheNS, title={The next simplest hyperbolic knots}, author={Abhijit Champanerkar and Ilya Kofman and Eric D. Patterson}, journal={arXiv: Geometric Topology}, year={2003} }

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.

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## References

SHOWING 1-10 OF 22 REFERENCES

THE SIMPLEST HYPERBOLIC KNOTS

- Mathematics
- 1999

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

- MathematicsComputers and Mathematics
- 1989

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

- Mathematics
- 2003

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

- Mathematics
- 2000

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

- Mathematics
- 1999

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

- Mathematics
- 1998

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

- Mathematics
- 1999

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

- Mathematics
- 1999

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

- MathematicsSpringer US
- 1989

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