---
date: 2021-11-29
tags: [ web/quick-notes ]
---

# 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 $f: V\tensorpower{}{n} \to V\tensorpower{}{m}$ 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 $H$:

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![](figures/2021-11-29_15-21-20.png)

![](figures/2021-11-29_15-24-10.png)

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

- Involutory: $s^2=\id$, 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:

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![](figures/2021-11-29_15-41-03.png)

- 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 $\inner{\wait}{\wait}:H_a\tensor H_b\to k$ 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 $X$.