We refer to the Acknowledgment page for contributors and methodology of computation.
Prime knots of up to 19 crossings:
The following folders contain the first prime knots of up to 19 crossings, with basic information:
- knot names [name] and signatures [knot_sig],
- alternating/non-alternating [alternating], torus/non-torus [torus], satellite/non-satellite [satellite], hyperbolic/non-hyperbolic [hyperbolic], with value in 0/1,
Notice
The knots diagrams have been taken from the Regina website where more information and representations of knot diagrams are available (DT code, DT name, topological structure). We use the same names for the knot diagrams. The knot census has been constituted thanks to the effort of many mathematicians and computer scientist: see Acknowledgment.Files have been divided to contain at most 10,000,000 knots.
Jones polynomials of knots:
The following folders contain the Jones polynomials of the first prime knots with up to 19 crossings, with:
- knot names [name],
- Jones polynomial [jones_poly], encoded as a sparse list of monomials “i a j b …” standing for “a X^i + b X^j …”
Notice
The Jones polynomials follow the convention of “Bar Nathan – On Khovanov’s categorification of the Jones polynomial“
HOMFLY-PT polynomials of knots:
The following folders contain the HOMFLY-PT polynomials of the first prime knots with up to 19 crossings, with:
- knot names [name],
- HOMFLY-PT polynomial [homfly_poly], encoded as a sparse list of monomials “i j a k l b …” standing for “a L^i M^j + b L^k M^l …”
Notice
The HOMFLY-PT polynomials are given with variables L and M.Hyperbolic volumes of hyperbolic knots
The following files contains the hyperbolic volumes of hyperbolic knots (separated into alternating and non-alternating):
- the columns contain respectively the name of the knot, the signature of a minimal diagram, a numerical approximation of the hyperbolic volume, and the iso-signature of the canonical triangulation.
Notice
The hyperbolic volume have been computed with the software SnapPy, in its exact calculation mode within Sage.For some knots, calculation failed ; we indicate them in the “errors” files.
Closed Orientable 3-Manifolds
The following contains the first prime closed orientable 3-manifolds, triangulable with at most 11 tetrahedra. It contains:
- 3-manifold names [name],
- the triangulation signature [tri_sig] of a minimal triangulation.
Notice
The census has been taken from the Regina website where more information and representations are available. We use the same names for the 3-manifolds. The census has been constituted thanks to the effort of many mathematicians and computer scientist: see Acknowledgment.Turaev-Viro Invariants of closed 3-manifolds – Numerical Approximation
Numerical approximation of the Turaev-Viro invariants (q=2) for the first prime 3-manfiolds.
Notice
These TV invariants are computed with multi-precision arithmetic, and the value are within 10^-5 of the exact value. They however come with no guarantee.The number of Turaev-Viro invariants available varies depending on the 3-manifolds, as they are exponentially harder to compute when the number of tetrahedra grows.
Turaev-Viro Invariants of closed 3-manifolds – Exact, TV_r,q
Exact computation of the Turaev-Viro invariants of the first prime closed 3-manifolds.
- presented with 3-manifold name [name] and Turaev-Viro invariant [TV_r,q].
Notice
The Turaev-Viro invariant is presented as an element of a cyclotomic field. It is represented as a polynomial with rational coefficients and of degree < r, with the format “i p q j s t …” standing for “p/q * X^i + s/t * X^j …”.We only store non-zero coefficients. In particular, “nothing” indicate the 0 polynomial.
- Turaev-Viro TV_3,1 of closed orientable 3-manifolds
- Turaev-Viro TV_3,2 of closed orientable 3-manifolds
- Turaev-Viro TV_4 of closed orientable 3-manifolds
- Turaev-Viro TV_5,1 of closed orientable 3-manifolds
- Turaev-Viro TV_5,2 of closed orientable 3-manifolds
- Turaev-Viro TV_6 of closed orientable 3-manifolds
- Turaev-Viro TV_7,1 of closed orientable 3-manifolds
- Turaev-Viro TV_7,2 of closed orientable 3-manifolds
- Turaev-Viro TV_8 of closed orientable 3-manifolds
- Turaev-Viro TV_9,1 of closed orientable 3-manifolds
- Turaev-Viro TV_9,2 of closed orientable 3-manifolds
- Turaev-Viro TV_10 of closed orientable 3-manifolds
- Turaev-Viro TV_11,1 of closed orientable 3-manifolds
- Turaev-Viro TV_11,2 of closed orientable 3-manifolds
- Turaev-Viro TV_12 of closed orientable 3-manifolds
- Turaev-Viro TV_13,1 of closed orientable 3-manifolds
- Turaev-Viro TV_13,2 of closed orientable 3-manifolds
to appear soon…
