Tetrahedral Census Name

The complement of a hyperbolic knot can be described as the union of ideal tetrahedra. An enumeration of hyperbolic knots based on the number of tetrahedra required has been undertaken. The numbering scheme is based first on the number of required tetrahedra and then on volumes. If this is not sufficient, geodesic lengths are used to break ties.

Initial data for knots with 6 or fewer tetrahedra was taken from [1]. Initial data for knots with 7 tetrahedra was taken from [2]. Initial data for knots with 8 tetrahedra was taken from [3]. This was extended to 9 tetrahedra by Burton and Dunfield in [4, 5], in which any inexact computations that had been used in previous work were replaced with provably correct computations.

For each knot that is within the census, we give a 4-tuple. For example, for the knot 7_4 we give the result: [6,s648,K6_28,7a_6]. This is interpreted as follows. The "6" means that the complete hyperbolic structure on the complement of 7_4 is built from 6 (and no fewer) ideal tetrahedra. The "s648" means that among all knots and links in closed three-manifolds having complete hyperbolic complements that can built from 6 (and no fewer) ideal tetrahedra, this is the 648th that arises, ordered by volume. (The letter "s" relates to the convention m: <6, s: 6, v: 7, t: 8, o: 9.) (Note that a given knot complement can be arise as a knot complement in more than one 3-manifold.) The name "K6_28" means that this knot is the 28th one cusped 3-manifold that arises. The "7a_6" is the DT name of the knot. If the KnotInfo results table does not give a tetrahedral census knot name, it has been proven to not be in the census; that is, it does not have a hyperbolic complement that can be built with fewer than 10 ideal tetrahedra.

We thank Marc Kegel for providing much of the KnotInfo census data.


[1] Callahan, P. J., Dean, J. C., and Weeks, J. R., "The simplest hyperbolic knots".

[2] Champanerkar, A., Kofman, I., and Patterson, E., "The next simplest hyperbolic knots".

[3] Champanerkar, A., Kofman, I., Mullen, T., "The 500 simplest hyperbolic knots".

[4] Dunfield, N. M., "A census of exceptional Dehn fillings".

[5] Burton, B, "The cusped hyperbolic census is complete".

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