The linking number is the simplest link invariant given by Gauss; it is the first Gauss diagram formula expressed by one arrow among two circles. Proceeding the next stage, we study the second Gauss diagram formula consisting of two arrows among two circles. We call a function of this type the double linking number. There are two double linking numbers; one of them is an ordinary Vassiliev invariant and the other function is surprisingly sensitive to the necessity of the second Reidemeister moves though any one-component. Gauss diagram formula cannot detect the necessity.

Gauss diagram; links; Vassiliev invariant; Reidemeister moves; Östlund conjecture.

Received: September 26, 2023;  Accepted: November 2, 2023; Published: December 13, 2023

How to cite this article: Kamolphat Intawong and Noboru Ito, Double linking number, JP Journal of Geometry and Topology 29(2) (2023), 187-197. http://dx.doi.org/10.17654/0972415X23010

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

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