132 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
On Upper Bounds for Toroidal Mosaic Numbers
In this paper, we work to construct mosaic representations of knots on the
torus, rather than in the plane. This consists of a particular choice of the
ambient group, as well as different definitions of contiguous and suitably
connected. We present conditions under which mosaic numbers might decrease by
this projection, and present a tool to measure this reduction. We show that the
order of edge identification in construction of the torus sometimes yields
different resultant knots from a given mosaic when reversed. Additionally, in
the Appendix we give the catalog of all 2 by 2 torus mosaics.Comment: 10 pages, 111 figure
The Topology of Branching Universes
The purpose of this paper is to survey the possible topologies of branching
space-times, and, in particular, to refute the popular notion in the literature
that a branching space-time requires a non-Hausdorff topology
A matrix solution to pentagon equation with anticommuting variables
We construct a solution to pentagon equation with anticommuting variables
living on two-dimensional faces of tetrahedra. In this solution, matrix
coordinates are ascribed to tetrahedron vertices. As matrix multiplication is
noncommutative, this provides a "more quantum" topological field theory than in
our previous works
The Jones polynomial and graphs on surfaces
The Jones polynomial of an alternating link is a certain specialization of
the Tutte polynomial of the (planar) checkerboard graph associated to an
alternating projection of the link. The Bollobas-Riordan-Tutte polynomial
generalizes the Tutte polynomial of planar graphs to graphs that are embedded
in closed oriented surfaces of higher genus.
In this paper we show that the Jones polynomial of any link can be obtained
from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph
associated to a link projection. We give some applications of this approach.Comment: 19 pages, 9 figures, minor change
Are Causality Violations Undesirable?
Causality violations are typically seen as unrealistic and undesirable
features of a physical model. The following points out three reasons why
causality violations, which Bonnor and Steadman identified even in solutions to
the Einstein equation referring to ordinary laboratory situations, are not
necessarily undesirable. First, a space-time in which every causal curve can be
extended into a closed causal curve is singularity free--a necessary property
of a globally applicable physical theory. Second, a causality-violating
space-time exhibits a nontrivial topology--no closed timelike curve (CTC) can
be homotopic among CTCs to a point, or that point would not be causally well
behaved--and nontrivial topology has been explored as a model of particles.
Finally, if every causal curve in a given space-time passes through an event
horizon, a property which can be called "causal censorship", then that
space-time with event horizons excised would still be causally well behaved.Comment: Accepted in October 2008 by Foundations of Physics. Latex2e, 6 pages,
no figures. Presented at a seminar at the Universidad Nacional Autonoma de
Mexico. Version 2 was co-winner of the QMUL CTC Essay Priz
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
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