A calculus for branched spine of 3-manifolds

TL;DR

This work develops a calculus for branched spines of 3-manifolds by extending the Matveev-Piergallini framework to branched moves and bubbles, enabling a connected path between any two branched standard spines of a given oriented 3-manifold $M$. The core idea is to show that two branchings on a triangulation can be related by a finite sequence of branched $2\rightarrow 3$-moves, lune-moves, and bubble-moves, while keeping intermediate spines standard, thus establishing transit invariance for branching state-sums. The method combines Makovetsky’s positive $2\rightarrow 3$ moves with a constructive, two-step procedure: first aligning edge-branchings through sequences of branching-sensitive moves, then adjusting vertex configurations to exchange $R^+_*$ with $R^-_*$ and eliminating auxiliary singularities. The results provide a robust basis for state-sum quantum invariants based on branched triangulations and offer a pathway toward a calculus of branched skeleta for 3-manifolds.

Abstract

We establish a calculus for branched spines of 3-manifolds by means of branched Matveev-Piergallini moves and branched bubble-moves. We briefly indicate some of its possible applications in the study and definition of State-Sum Quantum Invariants.

Figures (13)

• Figure 1: The local models of a simple polyhedron.
• Figure 2: In this figure we show the basic moves for polyhedra. Note that both moves create a new region: the small disc entirely contained in the left part of the figures.
• Figure 3: In the left part of the figure we show the bubble-move; it can be interpreted as the gluing of a disc $D$ along its boundary to a simple closed curve $c$ contained in the interior of a region $Y$ and bounding a small disc $D'$. The result of the application of such a move to a standard polyhedron is not a standard polyhedron, so often one uses its standard version, shown in tyhe right part of the picture.
• Figure 4: How a branching allows a smoothing of the polyhedron: the regions are oriented using the right-hand rule, the upward direction and the orientation of the ambient $3$-manifold (coinciding with the standard one of the chart depicted here).
• Figure 5: In the upper part of this figure we show the three branched versions of the lune-move called "sliding"-moves. In the bottom part we show the version called "bumping"-move. The arrow on the left indicates the vertical direction.

Theorems & Definitions (7)

• Theorem 2.1: Matveev-Piergallini
• Definition 2.2: Branching condition
• Remark 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• Theorem 3.1