Torus links $T_{2s,2t}$ and $(s,t)$-log VOA

TL;DR

This paper uncovers a deep link between torus links and logarithmic vertex operator algebras, demonstrating connections with quantum invariants, modularity, and providing a new geometric approach to character computation.

Contribution

It establishes a novel relationship between torus links and logarithmic VOAs, linking their characters to quantum invariants and introducing a geometric method for character calculation.

Findings

The singlet character matches the Kashaev invariant at roots of unity.

The character exhibits quantum modularity properties.

The tail of the colored Jones polynomial corresponds to the VOA character.

Abstract

We reveal an intimate connection between the torus link $T_{2s,2t}$ and the logarithmic $(s,t)$ VOA. We show that the singlet character of $(s,t)$-log VOA at the root of unity coincides with the Kashaev invariant and that it has a property of the quantum modularity. Also shown is that the tail of the $N$-colored Jones polynomial gives the character. Furthermore we propose a geometric method to compute the character.