On a Poincaré polynomial from Khovanov homology and Vassiliev invariants
Abstract
We introduce a Poincaré polynomial with twovariable $t$ 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 nontrivial 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 $y^n$ by the replacement $q=e^y$ for the quantum parameter $q$ of a quantum knot invariant, and which distinguishes the above knots together with the unknot. The first formulation is our polynomial.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.05664
 Bibcode:
 2019arXiv190505664I
 Keywords:

 Mathematics  Geometric Topology;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Quantum Algebra
 EPrint:
 8 pages, 2 figures, a Mathematica file is attached as ancillary file