Additional Gradings in Khovanov Homology

TL;DR

This paper introduces new gradings to the Khovanov complex, creating refined invariants for knots with additional structures, which enhance the ability to estimate knot characteristics and relate to existing homology theories.

Contribution

It constructs new gradings for Khovanov homology applicable to virtual knots and other structures, leading to sharper knot invariants and a spectral sequence linking to classical Khovanov homology.

Findings

New gradings lead to sharper estimates of knot invariants.

Constructed a spectral sequence from new homology to classical Khovanov homology.

Unified framework for gradings compatible with Frobenius extensions.

Abstract

The main goal of the present paper is to construct new invariants of knots with additional structure by adding new gradings to the Khovanov complex. The ideas given below work in the case of virtual knots, closed braids and some other cases of knots with additional structure. The source of our additional grading may be topological or combinatorial; it is axiomatised for many partial cases. As a byproduct, this leads to a complex which in some cases coincides (up to grading renormalisation) with the usual Khovanov complex and in some other cases with the Lee-Rasmussen complex. The grading we are going to construct behaves well with respect to some generalisations of the Khovanov homology, e.g., Frobenius extensions. These new homology theories give sharper estimates for some knot characteristics, such as minimal crossing number, atom genus, slice genus, etc. Our gradings generate a natural filtration on the usual Khovanov complex. There exists a spectral sequence starting with our homology and converging to the usual Khovanov homology.