topological graph theory In mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces. It also studies immersions of graphs. Embedding a graph in ...

, a ribbon graph is a way to represent

graph embedding In topological graph theory, an embedding (also spelled imbedding) of a Graph (discrete mathematics), graph G on a surface (mathematics), surface \Sigma is a representation of G on \Sigma in which points of \Sigma are associated with graph the ...

s, equivalent in power to signed

rotation system In combinatorial mathematics, rotation systems (also called combinatorial embeddings or combinatorial maps) encode embeddings of graphs onto orientable surfaces by describing the circular ordering of a graph's edges around each vertex. A more for ...

s or

graph-encoded map In topological graph theory, a graph-encoded map or gem is a method of encoding a cellular embedding of a graph using a different graph with four vertices per edge of the original graph. It is the topological analogue of runcination, a geometric ...

s. It is convenient for visualizations of embeddings, because it can represent unoriented

surfaces A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. Surface or surfaces may also refer to: Mathematics *Surface (mathematics), a generalization of a plane which needs not be flat *Surf ...

without self-intersections (unlike embeddings of the whole surface into three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

) and because it omits the parts of the surface that are far away from the graph, allowing holes through which the rest of the embedding can be seen. Ribbon graphs are also called fat graphs.

Definition

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).

Embeddings

A ribbon graph representation may be obtained from an embedding of a graph onto a surface (and a

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...

on the surface) by choosing a sufficiently small number

\epsilon

, and representing each vertex and edge by their

\epsilon

neighborhoods A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...

in the surface. For small values of

\epsilon

, the edge rectangles become long and thin like

ribbon A ribbon or riband is a thin band of material, typically cloth but also plastic or sometimes metal, used primarily as decorative binding and tying. Cloth ribbons are made of natural materials such as silk, cotton, and jute and of synthetic mater ...

s, giving the name to the representation. 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. 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 ''band decomposition''., 1.1.5 Band Decompositions, pp. 7–8. The surface onto which the graph is embedded may be determined by whether it is

orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...

(true if any cycle in the graph has an even number of twists) and by its

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...

. The embeddings that can be represented by ribbon graphs are the ones in which a graph is embedded onto a 2-

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

(without boundary) and in which each face of the embedding is a topological disk.

Equivalence

Two ribbon graph representations are said to be equivalent (and define

homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

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 into the other within 3d space: this notion of equivalence considers only the intrinsic topology of the representation, and not how it is embedded. However, ribbon graphs are also applied in

knot theory In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...

, and in this application weaker notions of equivalence that take into account the 3d embedding may also be used.

References

{{reflist, refs= {{citation, title=Structural Analysis of Complex Networks, first=Matthias, last=Dehmer, publisher=Springer, year=2010, isbn=9780817647896, page=267, url=https://books.google.com/books?id=0ymYdW1rtBUC&pg=PA267 {{citation, title=Graphs on Surfaces: Dualities, Polynomials, and Knots, series=SpringerBriefs in Mathematics, first1=Joanna A., last1=Ellis-Monaghan, author1-link=Jo Ellis-Monaghan, first2=Iain, last2=Moffatt, publisher=Springer, year=2013, isbn=9781461469711, contribution=1.1.4 Ribbon Graphs, pages=5–7, url=https://books.google.com/books?id=Y-l-KTIku40C&pg=PA5 {{citation , last = Dijkgraaf , first = Robbert , editor1-last = Fröhlich , editor1-first = J. , editor2-last = 't Hooft , editor2-first = G. , editor3-last = Jaffe , editor3-first = A. , editor4-last = Mack , editor4-first = G. , editor5-last = Mitter , editor5-first = P. K. , editor6-last = Stora , editor6-first = R. , arxiv = hep-th/9201003 , contribution = Intersection theory, integrable hierarchies and topological field theory , location = New York , mr = 1204453 , pages = 95–158 , publisher = Plenum , series = NATO Advanced Science Institutes Series B: Physics , title = New Symmetry Principles in Quantum Field Theory: Proceedings of the NATO Advanced Study Institute held in Cargèse, July 16–27, 1991 , volume = 295 , year = 1992 {{citation, title=Theta Functions and Knots, first=Răzvan, last=Gelca, publisher=World Scientific, year=2014, isbn=9789814520584, page=289, url=https://books.google.com/books?id=waK6CgAAQBAJ&pg=PA289 Topological graph theory