Drinfel'd doubles of the $n$-rank Taft algebras and a generalization of the Jones polynomial

TL;DR

This paper explores the Drinfel'd double of n-rank Taft algebras, describing their structure and simple modules, and introduces a generalized knot invariant that extends the Jones polynomial to higher ranks.

Contribution

It provides a detailed description of the Drinfel'd double of n-rank Taft algebras and develops a new family of knot invariants generalizing the Jones polynomial.

Findings

Recover the Jones polynomial at rank 1

Obtain a specialization of the Dubrovnik polynomial at rank 2

Construct a composite invariant for higher ranks

Abstract

In the paper, we describe the Drinfel'd double structure of the $n$-rank Taft algebra and all of its simple modules, and then endow its $R$-matrices with some application to knot invariants. The knot invariants we get is a generalization of the Jones polynomial, in particular, it recovers the Jones polynomial in rank $1$ case, while in rank $2$ case, it is the one-parameter specialization of the two-parameter unframed Dubrovnik polynomial, and in higher rank case it is the composite ($n$-power) of the Jones polynomial.