Complex

For other uses, see Complex (disambiguation).

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

Definition

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

Edge colored graph

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 $A(i)$ 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 $i \neq j$ then the intersection $A(i)\cup A(j)$ is empty.
• The graph $(V,A)$ is connected.

Permutation sequence

An n-complex is a set $\Omega$ and a sequence of permutations on $\Omega$, $A : \{1,\dots,n\} \rightarrow \mathrm{Aut}(\Omega)$ such that:

• For every $A(i)$ in A, $A(i)\circ A(i)=\mathrm{id}$
• For every $A(i)$ in A, there is no element of $\Omega$ which is a fixed point of $A(i)$.
• If $i \neq j$ then for any element $x\in\Omega$, $A(i)(x)\neq A(j)(x)$.
• The group generated by the permutations in A is transitive on $\Omega$.

Concepts

Elements

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 $A(j)$ where $i\neq j$.

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

Incidence

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

Bibliography

• 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.