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Table of contents
Introduction
1 Hopf superalgebras
1.1 Basic notions and examples
1.1.1 Super vector spaces
1.1.2 Graphical notation
1.1.3 Hopf superalgebras
1.1.4 Examples
1.1.5 Bosonization of Hopf superalgebras
1.1.6 Involutivity and semisimplicity
1.2 Integrals and cointegrals
1.2.1 Definitions and examples
1.2.2 Unimodularity
1.3 Hopf G-algebras
1.3.1 Definitions and basic properties
1.3.2 The semidirect product case
2 Sutured 3-manifolds
2.1 Balanced sutured 3-manifolds
2.2 Heegaard diagrams
2.2.1 The Reidemeister-Singer theorem
2.3 Extended Heegaard diagrams
2.3.1 Cut systems of surfaces
2.3.2 Extended diagrams and extended moves
2.3.3 Dual curves
2.4 Homology orientations
3 Kuperberg invariants for sutured 3-manifolds
3.1 Kuperberg invariants of closed 3-manifolds
3.1.1 Tensors associated to Heegaard diagrams
3.1.2 The original construction of Kuperberg
3.1.3 Virelizier’s extension
3.2 Extending to sutured manifolds: the unimodular case
3.2.1 A direct Fox calculus-like formula
3.2.2 Examples of computation
3.2.3 Relation to Virelizier’s extension
3.2.4 Some lemmas
3.2.5 Proof of invariance, special case
3.2.6 The disconnected case
3.3 Extending to sutured manifolds: non-unimodular case
3.3.1 Spinc structures and multipoints
3.3.2 Multipoints and basepoints
3.3.3 Normalizing Z via Spinc
3.3.4 Proof of invariance
3.3.5 The disconnected case with Spinc
3.4 Twisted Kuperberg polynomials
4 Recovering Reidemeister torsion
4.1 Basics of Reidemeister torsion
4.1.1 Algebraic torsion
4.1.2 Twisted Reidemeister torsion
4.1.3 Twisted Alexander polynomials
4.2 Twisted torsion of sutured manifolds
4.2.1 Twisted torsion from a Heegaard diagram
4.2.2 Twisted torsion for link complements
4.3 Reidemeister torsion from Hopf algebra theory
4.3.1 Lemmas on exterior algebras
4.3.2 Proof of Theorem 2
4.3.3 Particular cases of Theorem 2
