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Non-semisimple invariants using trisections and Hopf algebras (Julian Chaidez)

  • How most quantum invariants go: the inputs are

    • Noncommutative algebraic data, e.g. a quantum group or fusion category, and
    • A diagram for a topological object, e.g. a cell decomposition.
  • Then apply a combinatorial state-sum process and show it is diagram-independent.

  • Examples:

    • Knot polynomials: a Laurent algebra and a knot diagram to produce a polynomial.
    • Digraph Witten invariants: a finite group and a surgery diagram to produce numerical invariants.
    • Crane-Yetter: a fusion category (or an extension) and a framed triangulation of a 4-manifold to produce a number.
    • Kuperberg: a Hopf algebra and a Heegard diagram of a 3-manifold to produce a number.
    • Trisections: an involutary Hopf triple of 3 Hopf algebras and a trisection diagram for a 4-manifold to produce a number.
  • Why do this? The trisection invariant recovers e.g. Crane-Yetter in some cases, and is suspected to be more sensitive to diffeomorphism types.

  • Tensor diagrams: if \(f: V{ {}^{ \scriptstyle\otimes_{}^{n} } } \to V{ {}^{ \scriptstyle\otimes_{}^{m} } }\) can be written as a node in a graph with \(m\) incoming edges and \(n\) outgoing edges.

    • Composition is plugging an output of \(f\) into an input of \(g\), tensoring is vertically stacking.
  • A Hopf algebra \(H\):

  • Other tensors that exist for Hopf algebra: right integrals/cointegrals, phase ??

  • Involutory: \(s^2=\operatorname{id}\), similar to semisimple. Relatively boring for these types of invariants, but the easiest setting.

  • Balanced: slightly weaker and more general, more interesting things possible.

  • Balanced invariants for 3-manifolds: take a Heegaard diagram \((\Sigma, \alpha, \beta)\) with a singular combing: a singular vector field with one singularity on each curve and on one base point for \(\Sigma\).

    • Require index 1 on the blue/red curve singularities, flow out of singularities along curves and into singularities away from the curves.
  • Theorem: singular combings on \(\Sigma\) determine combings on \(Y\), i.e. a nonvanishing vector field.

  • Require that vector field is tangent to either red or blue curves at every intersection, and use this to define a rotation number:

  • Defining Kuperberg invariants: associate intersection points to a tensor diagram, combine them to close ends so it evaluates to a scalar.

  • Hopf triple: 3 Hopf algebras \(H_{ab}\) over a field \(k\) with pairings \({\left\langle {{-}},~{{-}} \right\rangle}:H_a\otimes H_b\to k\) for each pair \(a, b\).

  • Triple combing: 3 singular combings with a common index 0 or 2 base point which restrict to the same singular combing on overlaps.

  • Theorem: a triple combing determines a \(\spinc\) structure on \(X\).

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