We introduce a Poincaré polynomial in two-variables t and x for knots, derived from Khovanov homology, where the specialization (t, x) = (1, -1) is a Vassiliev invariant of order n. Since for every n, there exist non-trivial knots with the same value of the Vassiliev invariant of order n as that of the unknot, there has been no explicit formulation of a perturbative knot invariant which is a coefficient of yⁿ by the replacement q = e^y for the quantum parameter q of a quantum knot invariant, which distinguishes the above knots together with the unknot. The first formulation is our polynomial.

Jones polynomial, Vassiliev invariant, Khovanov polynomial.

Received: April 2, 2022; Accepted: May 12, 2022; Published: June 17, 2022

How to cite this article: Noboru Ito and Masaya Kameyama, On a Poincaré polynomial from Khovanov homology and Vassiliev invariants, JP Journal of Geometry and Topology 27 (2022), 33-48. http://dx.doi.org/10.17654/0972415X22003

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

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