64 research outputs found
From rubber bands to rational maps: A research report
This research report outlines work, partially joint with Jeremy Kahn and
Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal
surfaces with boundary. One one hand, this lets us tell when one rubber band
network is looser than another, and on the other hand tell when one conformal
surface embeds in another.
We apply this to give a new characterization of hyperbolic critically finite
rational maps among branched self-coverings of the sphere, by a positive
criterion: a branched covering is equivalent to a hyperbolic rational map if
and only if there is an elastic graph with a particular "self-embedding"
property. This complements the earlier negative criterion of W. Thurston.Comment: 52 pages, numerous figures. v2: New example
Perturbative 3-manifold invariants by cut-and-paste topology
We give a purely topological definition of the perturbative quantum
invariants of links and 3-manifolds associated with Chern-Simons field theory.
Our definition is as close as possible to one given by Kontsevich. We will also
establish some basic properties of these invariants, in particular that they
are universally finite type with respect to algebraically split surgery and
with respect to Torelli surgery. Torelli surgery is a mutual generalization of
blink surgery of Garoufalidis and Levine and clasper surgery of Habiro.Comment: 18 pages, many figures. The important change in this version is an
improved blowup construction. Also 20-30 typos have been correcte
Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree
Given a link in the three-sphere, Ozsv\'ath and Szab\'o showed that there is
a spectral sequence starting at the Khovanov homology of the link and
converging to the Heegaard Floer homology of its branched double cover. The aim
of this paper is to explicitly calculate this spectral sequence in terms of
bordered Floer homology. There are two primary ingredients in this computation:
an explicit calculation of bimodules associated to Dehn twists, and a general
pairing theorem for polygons. The previous part (arXiv:1011.0499) focuses on
computing the bimodules; this part focuses on the pairing theorem for polygons,
in order to prove that the spectral sequence constructed in the previous part
agrees with the one constructed by Ozsv\'ath and Szab\'o.Comment: 85 pages, 19 figures, v3: Version to appear in Journal of Topolog
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