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In the other direction, from a ribbon graph one may find the faces of its corresponding embedding as the components of the boundary of the topological surface formed by the ribbon graph. One may recover the surface itself by gluing a topological disk to the ribbon graph along each boundary component.
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graph embeddings) if they are related to each other that a homeomorphism of the topological space formed by the union of the vertex disks and edge rectangles that preserves the identification of these features. Ribbon graph representations may be equivalent even if it is not possible to deform one
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In a ribbon graph representation, each vertex of a graph is represented by a topological disk, and each edge is represented by a topological rectangle with two opposite ends glued to the edges of vertex disks (possibly to the same disk as each other).
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The partition of the surface into vertex disks, edge disks, and face disks given by the ribbon graph and this gluing process is a different but related representation of the embedding called a
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A ribbon graph with one vertex (the yellow disk), three edges (two of them twisted), and one face. It represents an embedding of a graph with three self-loops onto the connected sum of three
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Dijkgraaf, Robbert (1992), "Intersection theory, integrable hierarchies and topological field theory", in Fröhlich, J.; 't Hooft, G.; Jaffe, A.; Mack, G.; Mitter, P. K.; Stora, R. (eds.),
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into the other within 3d space: this notion of equivalence considers only the intrinsic topology of the representation, and not how it is embedded.
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The embeddings that can be represented by ribbon graphs are the ones in which a graph is embedded onto a 2-
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A ribbon graph representation may be obtained from an embedding of a graph onto a surface (and a
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Two ribbon graph representations are said to be equivalent (and define
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on the surface) by choosing a sufficiently small number
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Graphs on
Surfaces: Dualities, Polynomials, and Knots
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