We introduce a Poincaré polynomial with two-variable $t$ 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 $y^n$ 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.
- Pub Date:
- May 2019
- Mathematics - Geometric Topology;
- High Energy Physics - Theory;
- Mathematical Physics;
- Mathematics - Quantum Algebra
- 8 pages, 2 figures, a Mathematica file is attached as ancillary file