Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion
We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra \(H\) with its automorphism group \(\text{Aut}(H)\). These are topological invariants of balanced sutured 3-manifolds endowed with a homomorphism of the fundamental group into \...
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Cornell University Library, arXiv.org
08.06.2021
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| Abstract | We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra \(H\) with its automorphism group \(\text{Aut}(H)\). These are topological invariants of balanced sutured 3-manifolds endowed with a homomorphism of the fundamental group into \(\text{Aut}(H)\) and possibly with a \(\text{Spin}^c\) structure and a homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if \(H\) is \(\mathbb{N}\)-graded, they can be extended in a canonical way to polynomial invariants. When \(H\) is an exterior algebra, we show that this invariant specializes to a refinement of the twisted relative Reidemeister torsion of sutured 3-manifolds. We also give an explanation of our Fox calculus formulas in terms of a particular Hopf group-algebra. |
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| AbstractList | We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra \(H\) with its automorphism group \(\text{Aut}(H)\). These are topological invariants of balanced sutured 3-manifolds endowed with a homomorphism of the fundamental group into \(\text{Aut}(H)\) and possibly with a \(\text{Spin}^c\) structure and a homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if \(H\) is \(\mathbb{N}\)-graded, they can be extended in a canonical way to polynomial invariants. When \(H\) is an exterior algebra, we show that this invariant specializes to a refinement of the twisted relative Reidemeister torsion of sutured 3-manifolds. We also give an explanation of our Fox calculus formulas in terms of a particular Hopf group-algebra. |
| Author | Daniel López Neumann |
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| Copyright | 2021. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| DOI | 10.48550/arxiv.1911.02925 |
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| Title | Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion |
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