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We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.

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 $\pm {\mu }_{11}^{\pm }$ (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 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.

Call i-hedrite any 4-valent n-vertex plane graph, whose faces are 2-, 3- and 4-gons only and p₂+p₃ = i. The edges of an i-hedrite, as of any Eulerian plane graph, are partitioned by its central circuits, i.e. those, which are obtained by starting with an edge and continuing at each vertex by the edge opposite the entering one. So, any i-hedrite is a projection of an alternating link, whose components correspond to its central circuits.

Call an i-hedrite irreducible, if it has no rail-road, i.e. a circuit of 4-gonal faces, in which every 4-gon is adjacent to two of its neighbors on opposite edges.

We present the list of all i-hedrites with at most 15 vertices. Examples of other results:

(i) All i-hedrites, which are not 3-connected, are identified.

(ii) Any irreducible i-hedrite has at most i − 2 central circuits.

(iii) All i-hedrites without self-intersecting central circuits are listed.

(iv) All symmetry group of i-hedrites are listed.