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Embedded surface invariants via the Broda-Petit construction

Ik Jae Lee, David N Yetter

TL;DR

The paper develops embedded surface invariants for 4-manifolds by extending Petit's dichromatic construction to capture surface data via a nested pair of ribbon fusion categories $B\subset C$ and banded-link diagrams. It establishes a functorial Kirby coloring framework, proving a knot-theoretic lemma that justifies quantum-dimension weighted coefficients, and defines a normalized invariant $I_{+}(W)$ that is invariant under Kirby moves under suitable nondegeneracy conditions. The generalized construction for $(0,1,2)$-handlebody–surface pairs uses a symmetric Frobenius algebra $F$ in $B$ and a left $F$-module $M$ in $C$, with precise band/unlink colorings and normalization; invariance hinges on $\,B$-transparency and the band slide/swim moves. Concrete group-theoretic examples with Vec$_G$ and Vec$_H$ demonstrate sensitivity to genus and knotting, including explicit computations for $S^{4}$, CP$^{2}$, and the spun trefoil, and yield insight into the scope and limitations of the invariant. The work opens multiple avenues for refinement, normalization, and possible non-semisimple extensions (CGHP23).

Abstract

We recall Petit's construction of "dichromatic" invariants of 4-manifolds computed from Kirby diagrams using a nested pair of ribbon fusion categories $ B \subset C $ as initial data. Along the way we prove a lemma that fits the use of formal linear combinations of simple objects with quantum dimensions a coefficients as in the constructions of Reshetikhin-Turaev, Broda, and Petit more firmly in the functorial framework favored by the authors. We then show that Hughes et al.'s banded-link presentations of surfaces embedded in 4-manifolds provide a means whereby Frobenius algebra in $B$ together with a suitable module over it lying in $C$, give rise to an invariant of a surface-4-manifold pair. We provide a class of examples of suitable initial data and compute sufficient examples to show the invariant is sensitive to both genus and knotting.

Embedded surface invariants via the Broda-Petit construction

TL;DR

The paper develops embedded surface invariants for 4-manifolds by extending Petit's dichromatic construction to capture surface data via a nested pair of ribbon fusion categories and banded-link diagrams. It establishes a functorial Kirby coloring framework, proving a knot-theoretic lemma that justifies quantum-dimension weighted coefficients, and defines a normalized invariant that is invariant under Kirby moves under suitable nondegeneracy conditions. The generalized construction for -handlebody–surface pairs uses a symmetric Frobenius algebra in and a left -module in , with precise band/unlink colorings and normalization; invariance hinges on -transparency and the band slide/swim moves. Concrete group-theoretic examples with Vec and Vec demonstrate sensitivity to genus and knotting, including explicit computations for , CP, and the spun trefoil, and yield insight into the scope and limitations of the invariant. The work opens multiple avenues for refinement, normalization, and possible non-semisimple extensions (CGHP23).

Abstract

We recall Petit's construction of "dichromatic" invariants of 4-manifolds computed from Kirby diagrams using a nested pair of ribbon fusion categories as initial data. Along the way we prove a lemma that fits the use of formal linear combinations of simple objects with quantum dimensions a coefficients as in the constructions of Reshetikhin-Turaev, Broda, and Petit more firmly in the functorial framework favored by the authors. We then show that Hughes et al.'s banded-link presentations of surfaces embedded in 4-manifolds provide a means whereby Frobenius algebra in together with a suitable module over it lying in , give rise to an invariant of a surface-4-manifold pair. We provide a class of examples of suitable initial data and compute sufficient examples to show the invariant is sensitive to both genus and knotting.
Paper Structure (5 sections, 8 theorems, 28 equations)

This paper contains 5 sections, 8 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. The Broda-Petit construction
  3. Generalized Broda-Petit invariants for 4D (0,1,2)-handlebody/surface pairs
  4. Examples from group theoretic fusion categories
  5. Directions for future research

Key Result

Proposition 2.10

In any fusion category, the family of maps ${\mathsf d}_X:X\rightarrow X$ which multiplies each direct summand of $X$ by its quantum dimension is independent of the choices decompositions into direct summands and forms a natural transformation.

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 15 more