365 research outputs found
A Transcendental Invariant of Pseudo-Anosov Maps
For each pseudo-Anosov map on surface , we will associate it with a
-submodule of , denoted by . is
defined by an interaction between the Thurston norm and dilatation of
pseudo-Anosov maps. We will develop a few nice properties of and
give a few examples to show that is a nontrivial invariant. These
nontrivial examples give an answer to a question asked by McMullen: the minimal
point of the restriction of the dilatation function on fibered face need not be
a rational point.Comment: 32 pages, 10 figures, abstract has been modified by following
suggestion from Curtis McMulle
Crosscap numbers and the Jones polynomial
We give sharp two-sided linear bounds of the crosscap number (non-orientable
genus) of alternating links in terms of their Jones polynomial. Our estimates
are often exact and we use them to calculate the crosscap numbers for several
infinite families of alternating links and for several alternating knots with
up to twelve crossings. We also discuss generalizations of our results for
classes of non-alternating links.Comment: 27 pages. Minor corrections and modifications. To appear in Advances
of Mathematic
Exceptional surgeries on alternating knots
We give a complete classification of exceptional surgeries on hyperbolic
alternating knots in the 3-sphere. As an appendix, we also show that the
Montesinos knots M (-1/2, 2/5, 1/(2q + 1)) with q at least 5 have no
non-trivial exceptional surgeries. This gives the final step in a complete
classification of exceptional surgery on arborescent knots.Comment: 30 pages, 19 figures. v2: recomputation performed via the newest
version of hikmot, v3: revised according to referees' comments, to appear in
Comm. Anal. Geo
A random tunnel number one 3-manifold does not fiber over the circle
We address the question: how common is it for a 3-manifold to fiber over the
circle? One motivation for considering this is to give insight into the fairly
inscrutable Virtual Fibration Conjecture. For the special class of 3-manifolds
with tunnel number one, we provide compelling theoretical and experimental
evidence that fibering is a very rare property. Indeed, in various precise
senses it happens with probability 0. Our main theorem is that this is true for
a measured lamination model of random tunnel number one 3-manifolds.
The first ingredient is an algorithm of K Brown which can decide if a given
tunnel number one 3-manifold fibers over the circle. Following the lead of
Agol, Hass and W Thurston, we implement Brown's algorithm very efficiently by
working in the context of train tracks/interval exchanges. To analyze the
resulting algorithm, we generalize work of Kerckhoff to understand the dynamics
of splitting sequences of complete genus 2 interval exchanges. Combining all of
this with a "magic splitting sequence" and work of Mirzakhani proves the main
theorem.
The 3-manifold situation contrasts markedly with random 2-generator 1-relator
groups; in particular, we show that such groups "fiber" with probability
strictly between 0 and 1.Comment: This is the version published by Geometry & Topology on 15 December
200
On the zeroes of the Alexander polynomial of a Lorenz knot
We show that the zeroes of the Alexander polynomial of a Lorenz knot all lie
in some annulus whose width depends explicitly on the genus and the braid index
of the considered knot.Comment: 30p., final version, to appear in Ann. Inst. Fourie
A SYMPLECTIC PROLEGOMENON
A symplectic manifold gives rise to a triangulated A∞-category, the derived Fukaya category, which encodes information on Lagrangian submanifolds and dynamics as probed by Floer cohomology. This survey aims to give some insight into what the Fukaya category is, where it comes from and what symplectic topologists want to do with it.This is the author accepted manuscript. The final version is available from the American Mathematical Society via http://dx.doi.org/10.1090/S0273-0979-2015-01477-
Modular functors, cohomological field theories and topological recursion
Given a topological modular functor in the sense of Walker
\cite{Walker}, we construct vector bundles over ,
whose Chern classes define semi-simple cohomological field theories. This
construction depends on a determination of the logarithm of the eigenvalues of
the Dehn twist and central element actions. We show that the intersection of
the Chern class with the -classes in is
computed by the topological recursion of \cite{EOFg}, for a local spectral
curve that we describe. In particular, we show how the Verlinde formula for the
dimensions is retrieved from the
topological recursion. We analyze the consequences of our result on two
examples: modular functors associated to a finite group (for which
enumerates certain -principle
bundles over a genus surface with boundary conditions specified by
), and the modular functor obtained from Wess-Zumino-Witten
conformal field theory associated to a simple, simply-connected Lie group
(for which is the Verlinde
bundle).Comment: 50 pages, 2 figures. v2: typos corrected and clarification about the
use of ordered pairs of points for glueing. v3: unitarity assumption waived +
discussion of families index interpretation of the correlation functions for
Wess-Zumino-Witten theorie
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