# On a Poincar\'e polynomial from Khovanov homology and Vassiliev invariants

## TL;DR

This paper introduces a new two-variable Poincaré polynomial derived from Khovanov homology that encodes Vassiliev invariants and aims to distinguish knots with identical lower-order invariants.

## Contribution

The paper presents the first explicit polynomial formulation that captures Vassiliev invariants from Khovanov homology and distinguishes knots sharing the same invariants as the unknot.

## Key findings

- Defines a two-variable Poincaré polynomial from Khovanov homology.
- Specialization yields Vassiliev invariants of order n.
- Provides a knot invariant capable of distinguishing certain knots from the unknot.

## Abstract

We introduce a Poincar\'{e} 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.