Seifert-Torres Type Formulas for the Alexander Polynomial from Quantum $\mathfrak{sl}_2$

TL;DR

This paper introduces a diagrammatic calculus for unrolled quantum rak{sl}_2 at a fourth root of unity, enabling algebraic proofs of Seifert-Torres formulas and linking quantum invariants to the Alexander polynomial.

Contribution

It provides a novel algebraic approach to Seifert-Torres formulas and connects quantum invariants with the multivariable Alexander polynomial using diagrammatic calculus.

Findings

Proved Seifert-Torres type formulas algebraically.

Derived a skein relation for n-cabled double crossings.

Showed quantum invariants determine the multivariable Alexander polynomial.

Abstract

We develop a diagrammatic calculus for representations of unrolled quantum $\mathfrak{sl}_2$ at a fourth root of unity. This allows us to prove Seifert-Torres type formulas for certain splice links using quantum algebraic methods, rather than topological methods. Other applications of this diagrammatic calculus given here are a skein relation for $n$-cabled double crossings and a simple proof that the quantum invariant associated with these representations determines the multivariable Alexander polynomial.