msmat - Mock Seifert Matrix Calculator

Background

The Gordon-Litherland linking form [1] of a virtual knot (knot in thickened surface) is the map

$$\mathscr{L}_F: H_1(F) \times H_1(F) \longrightarrow \mathbb{Z}$$

given by

$$\mathscr{L}_F(\alpha, \beta) = \ell k(\tau \alpha, \beta).$$

The mock Seifert matrix $A$ of a virtual knot is the Gram matrix of the Gordon-Litherland linking form with repsect to some basis for $H_1(F)$. Hence, representing the same Gordon-Litherland linking form defines equivalence classes of mock Seifert matrices under unimodular congurence, i.e. the existence of unimodular $P$ such that

$$A' = P^\top A P.$$

Features

This package computes the mock Seifert matrix of an alternating virtual knot, and can compute some invariants derived from it, namely:

• The determinant of the mock Seifert matrix, $\mathup{det}\ S$.
• The dimension of the mock Seifert matrix, $\mathup{dim}\ S$
• The Kobayashi invariant of the mock Seifert matrix, $\mathup{tr}(S^\top S^{-1})$. See [2].
• The mock Alexander polynomial of the mock Seifert matrix, $\mathup{det}(tS - S^\top)$. See [1].

Usage

Compute mock Seifert matrix:

python msmat.py <gauss code> <flags>

The gauss code must be alternating.

Flags:

• -i prints invariants.
• -v verbose.
• -vv very verbose.
• -s compute instead the symmetrisation of the mock Seifert matrix, correposnding to the Gordon-Litherland pairing $\mathscr{G}_F$ (yet to be implemented).

Compute mock Seifert matrix and invariants for many knots:

python tabulate.py <flags>

Sources from file knots.txt and outputs to file out.txt. Expects knots.txt to be of the form:

3.6 --hv g0 O1-U2-O3-U1-O2-U3-
3.7 --hv g1 O1-U2-O3+U1-O2-U3+
4.105 idhv g1 O1-U2-O3-U1-O4-U3-O2-U4-
4.106 --hv g1 O1-U2-O3-U1-O4+U3-O2-U4+
4.107 i--v g2 O1-U2-O3+U1-O4+U3+O2-U4+
4.108 ---- g0 O1-U2+O3+U1-O4-U3+O2+U4-

Flags:

• -s compute instead the symmetrisation of the mock Seifert matrix, corresponding to the Gordon-Litherland pairing $\mathscr{G}_F$.

References

[1] https://arxiv.org/abs/2301.05946

[2] https://arxiv.org/abs/1904.04397