2 research outputs found
Unicellular maps and filtrations of the mapping class group
This article first answers to questions about connectedness of a new family
of graphs on unicellular maps. Answering these questions goes through a
description of the mapping class group as surgeries on unicellular maps. We
also show how unicellular maps encode subgroups of the mapping group and
provide filtrations of the mapping class group. These facts add a layer on the
ubiquitous character of unicellular maps
Flip Paths Between Lattice Triangulations
Diagonal flip paths between triangulations have been studied in the
combinatorial setting for nearly a century. One application of flip paths to
Euclidean distance geometry and Moebius geometry is a recent, simple,
constructive proof by Connelly and Gortler of the Koebe-Andreev-Thurston circle
packing theorem that relies on the existence of a flip path between any two
triangulation graphs. More generally, length and other structural quantities on
minimum (length) flip paths are metrics on the space of triangulations. In the
geometric setting, finding a minimum flip path between two triangulations is
NP-complete. However, for two lattice triangulations, used to model electron
spin systems, Eppstein and Caputo et al. gave algorithms running in
time, where is the number of points in the point-set.
Their algorithms apply to constrained flip paths that ensure a set of
\emph{constraint} edges are present in every triangulation along the path. We
reformulate the problem and provide an algorithm that runs in
time. In fact, for a large, natural class of
inputs, the bound is tight, i.e., our algorithm runs in time linear in the
length of this output flip path. Our results rely on structural elucidation of
minimum flip paths. Specifically, for any two lattice triangulations, we use
Farey sequences to construct a partially-ordered sets of flips, called a
minimum flip \emph{plan}, whose linear-orderings are minimum flip paths between
them. To prove this, we characterize a minimum flip plan that starts from an
equilateral lattice triangulation - i.e., a lattice triangulation whose edges
are all unit-length - and \emph{forces a point-pair to become an edge}. To the
best of our knowledge, our results are the first to exploit Farey sequences for
elucidating the structure of flip paths between lattice triangulations.Comment: 24 pages (33 with appendices), 8 figure