15 research outputs found
Distance and intersection number in the curve graph of a surface
In this work, we study the cellular decomposition of induced by a filling
pair of curves and , , and its connection
to the distance function in the curve graph of a closed orientable
surface of genus . Efficient geodesics were introduced by the first
author in joint work with Margalit and Menasco in 2016, giving an algorithm
that begins with a pair of non-separating filling curves that determine
vertices in the curve graph of a closed orientable surface and
computing from them a finite set of {\it efficient} geodesics. We extend the
tools of efficient geodesics to study the relationship between distance
, intersection number , and . The main result is
the development and analysis of particular configurations of rectangles in
called \textit{spirals}. We are able to show that, in some
special cases, the efficient geodesic algorithm can be used to build an
algorithm that reduces while preserving . At the end of the
paper, we note a connection of our work to the notion of extending geodesics.Comment: 20 pages, 17 figures. Changes: A key lemma (Lemma 5.6) was revised to
be more precise, an irrelevant proposition (Proposition 2.1) and example were
removed, unnecessary background material was taken out, some of the
definitions and cited results were clarified (including added figures,) and
Proposition 5.7 and Theorem 5.8 have been merged into a single theorem,
Theorem 4.
Criticality for the Gehring link problem
In 1974, Gehring posed the problem of minimizing the length of two linked
curves separated by unit distance. This constraint can be viewed as a measure
of thickness for links, and the ratio of length over thickness as the
ropelength. In this paper we refine Gehring's problem to deal with links in a
fixed link-homotopy class: we prove ropelength minimizers exist and introduce a
theory of ropelength criticality.
Our balance criterion is a set of necessary and sufficient conditions for
criticality, based on a strengthened, infinite-dimensional version of the
Kuhn--Tucker theorem. We use this to prove that every critical link is C^1 with
finite total curvature. The balance criterion also allows us to explicitly
describe critical configurations (and presumed minimizers) for many links
including the Borromean rings. We also exhibit a surprising critical
configuration for two clasped ropes: near their tips the curvature is unbounded
and a small gap appears between the two components. These examples reveal the
depth and richness hidden in Gehring's problem and our natural extension.Comment: This is the version published by Geometry & Topology on 14 November
200