Closed oriented 3-manifolds are subtle equivalence classes of plane graphs

TL;DR

This paper establishes a correspondence between closed oriented 3-manifolds and classes of plane graphs called blinks, using a finite set of local moves, simplifying the understanding of 3-manifold topology through combinatorial graph transformations.

Contribution

It introduces a new combinatorial framework linking 3-manifolds to blinks and reduces the complexity of moves needed to relate equivalent manifolds, enhancing topological and combinatorial understanding.

Findings

Blinks classify closed oriented 3-manifolds via local moves.

A simplified move set with 36 moves on 36 coins is sufficient.

Residual fraction links (flinks) generalize blackboard-framed links.

Abstract

A {\em blink} is a plane graph with an arbitrary bipartition of its edges. As a consequence of a recent result of Martelli, I show that the homeomorphisms classes of closed oriented 3-manifolds are in 1-1 correspondence with specific classes of blinks. In these classes, two blinks are equivalent if they are linked by a finite sequence of local moves, where each one appears in a concrete list of 64 moves: they organize in 8 types, each being essentially the same move on 8 simply related configurations. The size of the list can be substantially decreased at the cost of loosing symmetry, just by keeping a very simple move type, the {\em ribbon moves} denoted $\pm \mu_{11}^{\pm}$ (which are in principle redundant). The inclusion of $\pm \mu_{11}^{\pm}$ implies that all the moves corresponding to plane duality (the starred moves), except for $\mu_{20}^\star$ and $\mu_{02}^\star$, are redundant and the coin calculus is reduced to 36 moves on 36 coins. A {\em residual fraction link} or a {\em flink}, is a new object which generalizes {\em blackboard-framed link}. It plays an important role in this work. I try to make the topological exposition as complete as possible: about half of the exposition deals with the topological preliminaries. The objective is to make it easier for the combinatorially oriented readers to understand the paper. It is in the aegis of this work to find new important connections between 3-manifolds and plane graphs.