The knot invariant associated to two-parameter quantum algebras

TL;DR

This paper constructs a new quantum knot invariant using two-parameter quantum algebras and a monoidal functor from tangles to modules, expanding the toolkit for knot theory with algebraic methods.

Contribution

It introduces a novel quantum knot invariant derived from the two-parameter quantum algebra $U_{v,t}$ using a skew-Hopf pairing and a monoidal functor from tangles to modules.

Findings

Constructed the $ ext{R}$-matrix for $U_{v,t}$.

Developed a monoidal functor from tangle category to module category.

Derived a quantum knot invariant for tangles of type $(n,n)$.

Abstract

Using the skew-Hopf pairing, we obtain $\mathcal{R}$-matrix for the two-parameter quantum algebra $U_{v,t}$. We further construct a strict monoidal functor $\mathcal{T}$ from the tangle category $(\mathrm{OTa},\otimes, \emptyset)$ to the category $(\mathrm{Mod},\otimes, \mathbb{Q}(v,t))$ of $U_{v,t}$-modules . As a consequence, the quantum knot invariant of the tangle $L$ of type $(n,n)$ is obtained by the action of $\mathcal{T}$ on the closure $\tilde{L}$ of $L$.