Topological polymers have various topological types, and they are expressed by graphs. However, for the Jones polynomial, we have a difficulty to compute it; computational time grows exponentially with respect to the crossing number. The simplest Vassiliev invariant is the linking number and thus we seek a next simple one as the Milnor’s triple linking number. In this paper, we introduce simple Gauss diagram formulas of Vassiliev invariants of Milnor type. These are non-torsion valued, whereas the base-point-free Milnor’s triple linking number is usually torsion-valued.

Gauss diagram, Milnor invariant, links, Vassiliev invariant.

Received: May 9, 2022; Accepted: June 14, 2022; Published: July 7, 2022

How to cite this article: Kamolphat Intawong and Noboru Ito, Variations of Milnor’s triple linking number, JP Journal of Geometry and Topology 27 (2022), 67-75. http://dx.doi.org/10.17654/0972415X22005

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] Tetsuo Deguchi, On numerical applications of the Vassiliev invariants to computational problems in physics, Proceedings of the Conference on Quantum Topology (K. S. Manhattan, 1993), World Sci. Publ., River Edge, NJ, 1994, pp. 87-98.
[2] Noboru Ito, Space of chord diagrams on spherical curves, Internat. J. Math. 30(12) (2019), 25, Paper No. 1950060.
[3] Noboru Ito, A triple coproduct of curves and knots, arXiv, 2022.
[4] Noboru Ito and Natsumi Oyamaguchi, Gauss diagram formulas of Vassiliev invariants of 2-bouquet graphs, Topology Appl. 290 (2021), 13, Paper No. 107580.
[5] Olof-Petter Östlund, Invariants of knot diagrams and diagrammatic knot invariants, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)-Uppsala Universitet (Sweden), 2001.
[6] Olof-Petter Östlund, A diagrammatic approach to link invariants of finite degree, Math. Scand. 94(2) (2004), 295-319.
[7] Michael Polyak and Oleg Viro, Gauss diagram formulas for Vassiliev invariants, Internat. Math. Res. Notices (11) (1994), 8, 445ff.