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A complex is an edge-colored graph that generalizes the graph encoding of a map and the flag graph of a polytope.

Definition[edit | edit source]

There exist several equivalent ways to define the idea of a complex.[1]

Edge colored graph[edit | edit source]

The flag graph of a tetrahedron is a complex.

A complex is a connected properly edge colored simple graph.

To put it another way, a complex is a vertex set V, a color set C, and a function E from colors where yields the edges with color i such that:

  • For every color i the every vertex is incident on exactly 1 edge of color i.
  • If then the intersection is empty.
  • The graph is connected.

Permutation sequence[edit | edit source]

An n-complex is a set and a sequence of permutations on , such that:

  • For every in A,
  • For every in A, there is no element of which is a fixed point of .
  • If then for any element , .
  • The group generated by the permutations in A is transitive on .

Concepts[edit | edit source]

Elements[edit | edit source]

The edges, faces and vertices of tetrahedron flag graph shown above.

For a given i the elements of a complex of rank i correspond to the orbits of the group generated by permutations where .[1]

Alternatively the elements of rank i in a a complex correspond to the connected components in the graph with edges of color i removed.[1]

Incidence[edit | edit source]

Two elements are incident on each other if their connected components share a vertex.[2]

References[edit | edit source]

Bibliography[edit | edit source]

  • Garza-Vargas, Jorge; Hubard, Isabel (7 July 2018). "Polytopality of Maniplexes" (PDF). arXiv:1604.01164.
  • Wilson, Steve (2012). "Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators". Symmetry. doi:10.3390/sym4020265.