Closed, oriented, connected 3-manifolds are subtle equivalence classes of plane graphs
Abstract
A blink is a plane graph with an arbitrary bipartition of its edges. As a consequence of a recent result of Martelli, it is shown 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 are organized 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 ribbon moves denoted ±μ±11 (which are in principle redundant). The inclusion of ±μ±11 implies that all the moves corresponding to plane duality (the starred moves), except for μ⋆20 and μ⋆02, are redundant and the coin calculus is reduced to 36 moves on 36 coins. A residual fraction link or a flink is a new object which generalizes blackboard-framed link. It plays an important role in this work. It is in the aegis of this work to find new important connections between 3-manifolds and plane graphs.
