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

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.

25 pages, 19 figures

References in corpus (1)

• On Conway's potential function for colored links