On the censuses: The prime knots of up to 19 crossings have been computed by the joint effort of many topologists, notably Tait, Kirkman and Little (late 1800s, up to 10 crossings), Conway (1960s, up to 11 crossings), Dowker and Thistlethwaite and Week (1998, up to 16 crossings), Burton (2020, up to 19 crossings). We refer to “Burton – The Next 350 Million Knots” for a description of state-of-the-art techniques to construct such data base.
The closed 3-manifolds of up to 11 tetrahedra have been tabulated and made available by Ben Burton.
Clément Maria: Maintains the TOPAL database. CM has computed:
- the Jones polynomials of all prime knots of up to 19 crossings, using Regina,
- the HOMFLY-PT polynomials of all prime knots of up to 19 crossings, using Regina, implementing [B18],
- the Turaev-Viro invariants of the prime closed 3-manifolds triangulable with up to 11 tetrahedra, using Regina, implementing [BMS18], in conjonction with optimizations from [MR21]. So far, TV has been computed for r = 3 … 7.
Owen Rouillé: OR has computed:
- the hyperbolic volumes of the prime knots of up to 17 crossings, using Snappy and the optimizations of [MR22],
- the numerical approximations of Turaev-Viro invariants of closed 3-manifolds, using Regina, and the algorithms from [BMS18] and [MR21].
Bibliography:
- [MR22] Clément Maria, Owen Rouillé – Computing complete hyperbolic structures on cusped 3-manifolds – ESA 2022
- [M21] Clément Maria – Parameterized complexity of quantum knot invariants – SoCG 2021
- [MR21] Clément Maria, Owen Rouillé – Computation of large asymptotics of 3-manifold quantum invariants – SIAM-ALENEX 21
- [BMS18] Benjamin A. Burton, Clément Maria, Jonathan Spreer – Algorithms and complexity for Turaev-Viro invariants – JACT 2018
- [B18]Benjamin A. Burton – The HOMFLY-PT Polynomial is Fixed-Parameter Tractable – SoCG 2018
