The next simplest hyperbolic knots.

Abhijit Champanerkar

KNOTS in WASHINGTON XV (2nd Japan-USA Workshop in Knot Theory)
January 10-15, 2003
George Washington University and Johns Hopkins University
Washington, DC and Baltimore, MD, USA

Organizers
Kazuaki Kobayashi, Jozef H. Przytycki, Yongwu Rong, Shin-ichi Suzuki, Kouki Taniyama, Tatsuya Tsukamoto, Akira Yasuhara

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The next simplest hyperbolic knots.
by
Abhijit Champanerkar
Columbia University
Coauthors: Ilya Kofman, Eric Patterson

While crossing number is the standard notion of complexity for knots, it is hard to compute. The number of ideal tetrahedra required to construct the complement provides a natural alternative. Callahan, Dean and Weeks determined knot complements with 6 or fewer tetrahedra in SnapPea's census of cupsed hyperbolic 3-manifolds and explicitly described the corresponding knots. We extend their hyperbolic knot census and identify all knots whose complements have 7 tetrahedra. We obtain 129 knot complements out of 3552 orientable, cusped hyperbolic 3-manifolds with 7 tetrahedra. Many of these ``simple'' hyperbolic knots have high crossing number. We also compute their Jones polynomials. This is joint work with Ilya Kofman and Eric Patterson

Date received: November 7, 2002