counting and recursions
Last updated: 1 December 2014
Why do I care?
is the central character of a completed cycle
A fat graph
is an undirected graph with
a cyclic ordering of the edges that contain for each
each vertex is in edges.
A boundary component of is a sequence of distinct edges
such that for
is the successor of
is the successor of at
The genus and Euler characteristic of are
A metric fat graph is a pair where
is a fat graph and
The length of an edge is and the length of a boundary component
Let be a fat graph with boundary components labeled
Let be a fat graph.
so that is a directed graph with
where is the successor of
Then, in cycle notation
where are the boundary components of and
To consider the metric fat graph
insert vertices in and
Base case examples
labeled fat graphs and unlabeled fat graphs.
Relation to branched covers of
A branched cover of is a map
where is ....
Three permutations such that
specify a degree branched cover of
Notes and References
This is a typed copy of notes for the Algebra, Geometry and Topology seminar given on August 4, 2008 at the University of Melbourne.