This website lists the values of the algebraic genus [FL18] and the Taylor invariant [Tay77] for all prime knots with crossing number 12 or less. Details about how these values were computed are contained in the papers [LM19, FL19].
The Taylor invariant is the best lower bound for the topological (or smooth) slice genus that can be read off the S-equivalence class of the Seifert matrix (or equivalently, the isometry class of the Blanchfield pairing) [Tay77]. The algebraic genus is an upper bound for the topological slice genus; it may also be read off the S-equivalence class of the Seifert matrix. The algebraic genus of a knot K is in fact equal to its ℤ-slice genus [FL19], which is the minimal genus of a topological slice surface of K whose complement has fundamental group isomorphic to ℤ.
[FL18] [On classical upper bounds for slice genera] by P. Feller and L. Lewark, Selecta Math. 24 (2018), no. 5, 4885–4916. MR3874707, Zbl 1404.57008, arXiv: 1611.02679.
[Tay77] [On the genera of knots] by L. R. Taylor, In Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), volume 722 of Lecture Notes in Math., pages 144–154. Springer, Berlin, 1979. MR0547461, Zbl 0409.57021.
[FL19] [Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space] by P. Feller and L. Lewark, accepted in J. Differential Geom. arXiv:1905.08305.
[LM19] [On calculating the slice genera of 11- and 12-crossing knots] by L. Lewark and D. McCoy, Exp. Math. 28 (2019), no. 1, 81–94. MR3938580, Zbl 1414.57009, arXiv:1508.01098.
The table is available here:
The table is generated from the source file data.csv by running the script converter.py:
./converter.py data.csv > index.html
This script is a modified copy of a script written by Claudius Zibrowius for a similar online table.
While working on this program, Lukas Lewark was supported by the Emmy Noether Programme of the DFG, project number 412851057.