183 research outputs found
On Krebes' tangle
A genus-1 tangle G is an arc properly embedded in a standardly embedded solid
torus S in the 3-sphere. We say that a genus-1 tangle embeds in a knot K in S^3
if the tangle can be completed by adding an arc exterior to the solid torus to
form the knot K. We call K a closure of G. An obstruction to embedding a
genus-1 tangle G in a knot is given by torsion in the homology of branched
covers of S branched over G. We examine a particular example A of a genus-1
tangle, given by Krebes, and consider its two double-branched covers. Using
this homological obstruction, we show that any closure of A obtained via an arc
which passes through the hole of S an odd number of times must have determinant
divisible by three. A resulting corollary is that if A embeds in the unknot,
then the arc which completes A to the unknot must pass through the hole of S an
even number of times.Comment: 7 pages, 7 figures. v2: Minor changes made, typos corrected. v3:
Final version, accepted for publicatio
Abelian BF theory and Turaev-Viro invariant
The U(1) BF Quantum Field Theory is revisited in the light of
Deligne-Beilinson Cohomology. We show how the U(1) Chern-Simons partition
function is related to the BF one and how the latter on its turn coincides with
an abelian Turaev-Viro invariant. Significant differences compared to the
non-abelian case are highlighted.Comment: 47 pages and 6 figure
Additivity of Circular Width
We show that circular width is preserved under connected sum of knots for
some cases.Comment: 17 pages, 6 figure
Linked and knotted beams of light, conservation of helicity and the flow of null electromagnetic fields
Maxwell's equations allow for some remarkable solutions consisting of pulsed
beams of light which have linked and knotted field lines. The preservation of
the topological structure of the field lines in these solutions has previously
been ascribed to the fact that the electric and magnetic helicity, a measure of
the degree of linking and knotting between field lines, are conserved. Here we
show that the elegant evolution of the field is due to the stricter condition
that the electric and magnetic fields be everywhere orthogonal. The field lines
then satisfy a `frozen field' condition and evolve as if they were unbreakable
filaments embedded in a fluid. The preservation of the orthogonality of the
electric and magnetic field lines is guaranteed for null, shear-free fields
such as the ones considered here by a theorem of Robinson. We calculate the
flow field of a particular solution and find it to have the form of a Hopf
fibration moving at the speed of light in a direction opposite to the
propagation of the pulsed light beam, a familiar structure in this type of
solution. The difference between smooth evolution of individual field lines and
conservation of electric and magnetic helicity is illustrated by considering a
further example in which the helicities are conserved, but the field lines are
not everywhere orthogonal. The field line configuration at time t=0 corresponds
to a nested family of torus knots but unravels upon evolution
Supersymmetry, homology with twisted coefficients and n-dimensional knots
Let be any natural number. Let be any -dimensional knot in
. We define a supersymmetric quantum system for with the following
properties. We firstly construct a set of functional spaces (spaces of
fermionic \{resp. bosonic\} states) and a set of operators (supersymmetric
infinitesimal transformations) in an explicit way. Thus we obtain a set of the
Witten indexes for . Our Witten indexes are topological invariants for
-dimensional knots. Our Witten indexes are not zero in general. If is
equivalent to the trivial knot, all of our Witten indexes are zero. Our Witten
indexes restrict the Alexander polynomials of -knots. If one of our Witten
indexes for an -knot is nonzero, then one of the Alexander polynomials
of is nontrivial. Our Witten indexes are connected with homology with
twisted coefficients. Roughly speaking, our Witten indexes have path integral
representation by using a usual manner of supersymmetric theory.Comment: 10pages, no figure
On the Kauffman bracket skein module of the quaternionic manifold
We use recoupling theory to study the Kauffman bracket skein module of the
quaternionic manifold over Z[A,A^{-1}] localized by inverting all the
cyclotomic polynomials. We prove that the skein module is spanned by five
elements. Using the quantum invariants of these skein elements and the Z_2
homology of the manifold, we determine that they are linearly independent.Comment: corrected summation signs in figures 14, 15, 17. Other minor change
Lens Spaces and Handlebodies in 3D Quantum Gravity
We calculate partition functions for lens spaces L_{p,q} up to p=8 and for
genus 1 and 2 handlebodies H_1, H_2 in the Turaev-Viro framework. These can be
interpreted as transition amplitudes in 3D quantum gravity. In the case of lens
spaces L_{p,q} these are vacuum-to-vacuum amplitudes \O -> \O, whereas for
the 1- and 2-handlebodies H_1, H_2 they represent genuinely topological
transition amplitudes \O -> T^2 and \O -> T^2 # T^2, respectively.Comment: 14 pages, LaTeX, 5 figures, uses eps
Higher dimensional abelian Chern-Simons theories and their link invariants
The role played by Deligne-Beilinson cohomology in establishing the relation
between Chern-Simons theory and link invariants in dimensions higher than three
is investigated. Deligne-Beilinson cohomology classes provide a natural abelian
Chern-Simons action, non trivial only in dimensions , whose parameter
is quantized. The generalized Wilson -loops are observables of the
theory and their charges are quantized. The Chern-Simons action is then used to
compute invariants for links of -loops, first on closed
-manifolds through a novel geometric computation, then on
through an unconventional field theoretic computation.Comment: 40 page
Post Quantum Cryptography from Mutant Prime Knots
By resorting to basic features of topological knot theory we propose a
(classical) cryptographic protocol based on the `difficulty' of decomposing
complex knots generated as connected sums of prime knots and their mutants. The
scheme combines an asymmetric public key protocol with symmetric private ones
and is intrinsecally secure against quantum eavesdropper attacks.Comment: 14 pages, 5 figure
3-manifold invariants and periodicity of homology spheres
We show how the periodicity of a homology sphere is reflected in the
Reshetikhin-Turaev-Witten invariants of the manifold. These yield a criterion
for the periodicity of a homology sphere.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-34.abs.htm
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