ribbonor

Ribbonor is a theoretical construct in topology that generalizes the notion of ribbon graphs by attaching a twist datum to each edge. It is defined from a finite graph G = (V,E) together with a cyclic ordering of the half-edges at every vertex, as in fatgraphs, and a function t: E → {0,1} that records whether the corresponding edge carries a half-twist in its neighborhood.

To realize a ribbonor geometrically, each edge is replaced by a rectangular band, and at each vertex the adjacent bands are glued according to the prescribed cyclic order. An edge e with t(e)=0 is untwisted; with t(e)=1 the band is given a half-twist. The resulting thickened embedding yields a compact surface with boundary, whose boundary components depend on both the cyclic orders and the twists. When all t(e)=0 the construction recovers the ordinary ribbon graph. The twist data also determine non-orientable features of the surface.

Ribbonors are studied up to ribbon-preserving homeomorphisms of the ambient surface, i.e., homeomorphisms that preserve both the vertex cyclic orders and the edge twists. Invariants associated with a ribbonor include the genus of the thickened surface, the number of boundary components, and the total twist parity.

Applications of ribbonors appear in topological graph theory and related areas of low-dimensional topology. They provide a flexible framework for modeling embedded graphs with boundary, for encoding non-orientable structures, and for formulating state-sum constructions in skein theory and topological quantum field theories. See also ribbon graphs, fatgraphs, and skein modules. The concept has appeared in recent mathematical literature as a natural extension of established graph-embedding formalisms.