Spectral sequences of colored Jones polynomials, colored Rasmussen invariants and nanophrases

TL;DR

This paper introduces three spectral sequences related to colored Jones polynomials, linking them to Khovanov-type homologies, and explores their applications through functors between nanophrases and link categories.

Contribution

It presents new spectral sequences for colored Jones polynomials and establishes functors connecting nanophrases with link invariants, expanding categorification frameworks.

Findings

Derived spectral sequences from bicomplexes of colored Jones polynomials

Established a spectral sequence for colored Rasmussen invariants

Connected nanophrases with link invariants via functors

Abstract

We introduce three spectral sequences which give some expressions of colored Jones polynomials. Each spectral sequence contains a Khovanov-type homology groups. Two of them are derived from a bicomplex of the colored Jones polynomial. The other is the spectral sequence that deduces a colored Rasmussen invariant of links. We also introduce three functors between categories of nanophrases, generalizations of links, and obtain their applications using colored Jones polynomials and their categorifications.