# 2021-11-29

Tags: #web/quick-notes

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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 $$X$$.