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

Matthew Harper

doi:10.1016/j.topol.2022.108238

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arXiv · 1911.00646

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

Matthew Harper

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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.

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Matthew Harper. 2019-11-02. Seifert-Torres Type Formulas for the Alexander Polynomial from Quantum $\mathfrak{sl}_2$. https://doi.org/10.1016/j.topol.2022.108238

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