Band graph

In the mathematical field of graph theory is known graph in which each node is provided with a cyclic arrangement of the outgoing edges, as a band graph .

In topology , band graphs are useful for examining the topology of surfaces.

definition

A band graph: the arrows indicate the cyclic order of the edges.

For a graph denote the set of nodes, the set of edges and the set of directed edges, where ${\displaystyle (V,E)}$ ${\displaystyle V}$ ${\displaystyle E}$ ${\displaystyle {\overline {E}}\subset V\times V}$

${\displaystyle (v,w)\in {\overline {E}}\Leftrightarrow (w,v)\in {\overline {E}}\Leftrightarrow \left\{v,w\right\}\in E}$.

For each node we denote with ${\displaystyle v\in V}$

${\displaystyle St(v)=\left\{e=(v,w)\in {\overline {E}}\colon w\in V\right\}}$

the amount of outgoing directed edges. ${\displaystyle v}$

Definition : A band graph is a graph along with a cyclic arrangement of the directed edges from for each . ${\displaystyle (V,E)}$ ${\displaystyle St(v)}$ ${\displaystyle v\in V}$

That is, for each one has a permutation ${\displaystyle v\in V}$

${\displaystyle \sigma _{v}\colon St(v)\to St(v)}$,

so that for each one its orbit is completely below : ${\displaystyle x\in St(v)}$ ${\displaystyle \sigma _{v}}$ ${\displaystyle St(v)}$

${\displaystyle \left\{\sigma _{v}^{n}(x)\colon n\in \mathbb {Z} \right\}=St(v)}$.

Equivalently, one can request that there is a permutation

${\displaystyle \sigma \colon {\overline {E}}\to {\overline {E}}}$

are whose cycles correspond exactly to cyclical arrangements on the sets with . The relationship between the two equivalent definitions is given by the equation ${\displaystyle St(v)}$ ${\displaystyle v\in V}$

${\displaystyle \sigma (e)=\sigma _{v}(e)\Leftrightarrow e\in St(v)}$.

Assigned areas

A band graph can be assigned an area with a border by assigning a rectangle to each edge of the graph and a circular disk to each node and gluing the rectangles to the circular disks in accordance with the given cyclic order.

You can also assign a closed area to the band graph by gluing the edge components of the area constructed above with a circular disk.

This construction enables an elementary proof of the classification of the surfaces and it is useful in the investigation of the mapping class groups of surfaces.

Designate the category of connected band graphs in which every node is adjacent with at least 3 edges, then the geometric realization is weakly homotopy equivalent to the disjoint union of the classifying spaces of the mapping class groups for all surfaces: ${\displaystyle FatGraph_{c}^{3}}$ ${\displaystyle |FatGraph_{c}^{3}|}$

${\displaystyle |FatGraph_{c}^{3}|\sim \bigcup _{S}BMod(S)}$.

literature

• François Laborie : Lectures on Representations of Surface Groups . Zurich Lecture Notes in Advanced Mathematics (2013). ISBN 978-3-03719-127-9 (print), ISBN 978-3-03719-627-4 (online), doi : 10.4171 / 127 online (PDF)

Web links

• Ribbon Graph (nLab)

Individual evidence

1. ↑ Kevin Costello: A dual point of view on the ribbon graph decomposition of moduli space.
2. ↑ Maxim Kontsevich: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. (1992), no. 147, pp. 1-23.
3. ↑ Kiyoshi Igusa: Higher Franz Reidemeister torsion. IP Studies in Advanced Mathematics, American Mathematical Society, 2002.
4. ↑ K. Strebel: Quadratic Differentials Springer, Berlin 1984, MR86a: 30072.
5. ^ RC Penner: The decorated Teichmüller space of punctured surfaces. Commun. Math. Phys. 113 (2) (1987) pp. 299-339.