59 research outputs found
Invariants of genus 2 mutants
Pairs of genus 2 mutant knots can have different Homfly polynomials, for
example some 3-string satellites of Conway mutant pairs. We give examples which
have different Kauffman 3-variable polynomials, answering a question raised by
Dunfield et al in their study of genus 2 mutants. While pairs of genus 2 mutant
knots have the same Jones polynomial, given from the Homfly polynomial by
setting v=s^2, we give examples whose Homfly polynomials differ when v=s^3. We
also give examples which differ in a Vassiliev invariant of degree 7, in
contrast to satellites of Conway mutant knots.Comment: 16 pages, 20 figure
SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial
We clarify and refine the relation between the asymptotic behavior of the
colored Jones polynomial and Chern-Simons gauge theory with complex gauge group
SL(2,C). The precise comparison requires a careful understanding of some
delicate issues, such as normalization of the colored Jones polynomial and the
choice of polarization in Chern-Simons theory. Addressing these issues allows
us to go beyond the volume conjecture and to verify some predictions for the
behavior of the subleading terms in the asymptotic expansion of the colored
Jones polynomial.Comment: 15 pages, 7 figure
Quantum Knitting
We analyze the connections between the mathematical theory of knots and
quantum physics by addressing a number of algorithmic questions related to both
knots and braid groups.
Knots can be distinguished by means of `knot invariants', among which the
Jones polynomial plays a prominent role, since it can be associated with
observables in topological quantum field theory.
Although the problem of computing the Jones polynomial is intractable in the
framework of classical complexity theory, it has been recently recognized that
a quantum computer is capable of approximating it in an efficient way. The
quantum algorithms discussed here represent a breakthrough for quantum
computation, since approximating the Jones polynomial is actually a `universal
problem', namely the hardest problem that a quantum computer can efficiently
handle.Comment: 29 pages, 5 figures; to appear in Laser Journa
Average Structures of a Single Knotted Ring Polymer
Two types of average structures of a single knotted ring polymer are studied
by Brownian dynamics simulations. For a ring polymer with N segments, its
structure is represented by a 3N -dimensional conformation vector consisting of
the Cartesian coordinates of the segment positions relative to the center of
mass of the ring polymer. The average structure is given by the average
conformation vector, which is self-consistently defined as the average of the
conformation vectors obtained from a simulation each of which is rotated to
minimize its distance from the average conformation vector. From each
conformation vector sampled in a simulation, 2N conformation vectors are
generated by changing the numbering of the segments. Among the 2N conformation
vectors, the one closest to the average conformation vector is used for one
type of the average structure. The other type of the averages structure uses
all the conformation vectors generated from those sampled in a simulation. In
thecase of the former average structure, the knotted part of the average
structure is delocalized for small N and becomes localized as N is increased.
In the case of the latter average structure, the average structure changes from
a double loop structure for small N to a single loop structure for large N,
which indicates the localization-delocalization transition of the knotted part.Comment: 15 pages, 19 figures, uses jpsj2.cl
Homotopy types of stabilizers and orbits of Morse functions on surfaces
Let be a smooth compact surface, orientable or not, with boundary or
without it, either the real line or the circle , and
the group of diffeomorphisms of acting on by the rule
, where and .
Let be a Morse function and be the orbit of under this
action. We prove that for , and
except for few cases. In particular, is aspherical, provided so is .
Moreover, is an extension of a finitely generated free abelian
group with a (finite) subgroup of the group of automorphisms of the Reeb graph
of .
We also give a complete proof of the fact that the orbit is tame
Frechet submanifold of of finite codimension, and that the
projection is a principal locally trivial -fibration.Comment: 49 pages, 8 figures. This version includes the proof of the fact that
the orbits of a finite codimension of tame action of tame Lie group on tame
Frechet manifold is a tame Frechet manifold itsel
On the Quantization of the Chern-Simons Fields Theory on Curved Space-Times: the Coulomb Gauge Approach
We consider here the Chern-Simons field theory with gauge group SU(N) in the
presence of a gravitational background that describes a two-dimensional
expanding ``universe". Two special cases are treated here in detail: the
spatially flat {\it Robertson-Walker} space-time and the conformally static
space-times having a general closed and orientable Riemann surface as spatial
section. The propagator and the vertices are explicitely computed at the lowest
order in perturbation theory imposing the Coulomb gauge fixing.Comment: 15 pp., Preprint LMU-TPW 93-5, (Plain TeX + Harvmac
Quantum geometry and quantum algorithms
Motivated by algorithmic problems arising in quantum field theories whose
dynamical variables are geometric in nature, we provide a quantum algorithm
that efficiently approximates the colored Jones polynomial. The construction is
based on the complete solution of Chern-Simons topological quantum field theory
and its connection to Wess-Zumino-Witten conformal field theory. The colored
Jones polynomial is expressed as the expectation value of the evolution of the
q-deformed spin-network quantum automaton. A quantum circuit is constructed
capable of simulating the automaton and hence of computing such expectation
value. The latter is efficiently approximated using a standard sampling
procedure in quantum computation.Comment: Submitted to J. Phys. A: Math-Gen, for the special issue ``The
Quantum Universe'' in honor of G. C. Ghirard
A 2018 Horizon Scan of Emerging Issues for Global Conservation and Biological Diversity.
This is our ninth annual horizon scan to identify emerging issues that we believe could affect global biological diversity, natural capital and ecosystem services, and conservation efforts. Our diverse and international team, with expertise in horizon scanning, science communication, as well as conservation science, practice, and policy, reviewed 117 potential issues. We identified the 15 that may have the greatest positive or negative effects but are not yet well recognised by the global conservation community. Themes among these topics include new mechanisms driving the emergence and geographic expansion of diseases, innovative biotechnologies, reassessments of global change, and the development of strategic infrastructure to facilitate global economic priorities
A horizon scan of global conservation issues for 2014
This paper presents the output of our fifth annual horizon-scanning exercise, which aims to identify topics that increasingly may affect conservation of biological diversity, but have yet to be widely considered. A team of professional horizon scanners, researchers, practitioners, and a journalist identified 15 topics which were identified via an iterative, Delphi-like process. The 15 topics include a carbon market induced financial crash, rapid geographic expansion of macroalgal cultivation, genetic control of invasive species, probiotic therapy for amphibians, and an emerging snake fungal disease. © 2013 Elsevier Ltd
Promoting Essential Laminations
We show that every co--orientable taut foliation F of an orientable,
atoroidal 3-manifold admits a transverse essential lamination. If this
transverse lamination is a foliation G, the pair F,G are the unstable and
stable foliation respectively of an Anosov flow. Otherwise, F admits a pair of
transverse very full genuine laminations.
In the second case, M satisfies the weak geometrization conjecture - either
its fundamental group contains Z+Z or it is word-hyperbolic. Moreover, if M is
atoroidal, the mapping class group of M is finite, and any automorphism
homotopic to the identity is isotopic to the identity.Comment: 56 pages, 11 figures; version 3: final version, incorporates
referee's suggestion
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