1,877 research outputs found
On Kontsevich integral of torus knots
We study the unwheeled rational Kontsevich integral of torus knots. We give a
precise formula for these invariants up to loop degree 3 and show that they
appear as colorings of simple diagrams. We show that they behave under cyclic
branched coverings in a very simple way. Our proof is combinatorial: it uses
the results of Wheels and Wheelings and new decorations of diagrams.Comment: 10 pages, 4 figure
The skein module of torus knots complements
We compute the Kauffman skein module of the complement of torus knots in S^3.
Precisely, we show that these modules are isomorphic to the algebra of
Sl(2,C)-characters tensored with the ring of Laurent polynomials.Comment: 10 pages, 1 figur
A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes
In this paper, we introduce a discontinuous Finite Element formulation on
simplicial unstructured meshes for the study of free surface flows based on the
fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a
new class of asymptotically equivalent equations, which have a simplified
analytical structure, we consider a decoupling strategy: we approximate the
solutions of the classical shallow water equations supplemented with a source
term globally accounting for the non-hydrostatic effects and we show that this
source term can be computed through the resolution of scalar elliptic
second-order sub-problems. The assets of the proposed discrete formulation are:
(i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary
order of approximation in space, (iii) the exact preservation of the motionless
steady states, (iv) the preservation of the water height positivity, (v) a
simple way to enhance any numerical code based on the nonlinear shallow water
equations. The resulting numerical model is validated through several
benchmarks involving nonlinear wave transformations and run-up over complex
topographies
Knot state asymptotics I, AJ Conjecture and abelian representations
Consider the Chern-Simons topological quantum field theory with gauge group
SU(2) and level k. Given a knot in the 3-sphere, this theory associates to the
knot exterior an element in a vector space. We call this vector the knot state
and study its asymptotic properties when the level is large. The latter vector
space being isomorphic to the geometric quantization of the SU(2)-character
variety of the peripheral torus, the knot state may be viewed as a section
defined over this character variety. We first conjecture that the knot state
concentrates in the large level limit to the character variety of the knot.
This statement may be viewed as a real and smooth version of the AJ conjecture.
Our second conjecture says that the knot state in the neighborhood of abelian
representations is a Lagrangian state. Using microlocal techniques, we prove
these conjectures for the figure eight and torus knots. The proof is based on
q-difference relations for the colored Jones polynomial. We also provide a new
proof for the asymptotics of the Witten-Reshetikhin-Turaev invariant of the
lens spaces and a derivation of the Melvin-Morton-Rozansky theorem from the two
conjectures.Comment: 47 pages, 2 figure
Further New Records of Coleoptera and Other Insects from Wisconsin
Specimens of eleven different species of insects, representing seven separate families of Coleoptera, and one family each of Hemiptera, Lepidoptera, Diptera, and Hymenoptera, are herein reported as new to Wisconsin. These genera or species occur respectively within the following families: Leiodidae, Monotomidae, Cucujidae, Cryptophagidae, Ciidae, Tetratomidae, Curculionidae, Pentatomidae, Glyphipterigidae, Phoridae, and Pteromalidae. All but one of these insects were collected at or near the author’s residence (Dane County); the pentatomid was taken in northern Wisconsin (Oconto County). Three of the four non-coleopteran fauna are introduced species
Generating series and asymptotics of classical spin networks
We study classical spin networks with group SU(2). In the first part, using
gaussian integrals, we compute their generating series in the case where the
networks are equipped with holonomies; this generalizes Westbury's formula. In
the second part, we use an integral formula for the square of the spin network
and perform stationary phase approximation under some non-degeneracy
hypothesis. This gives a precise asymptotic behavior when the labels are
rescaled by a constant going to infinity.Comment: 33 pages, 3 figures; in version 2 added one reference and a comment
on the hypotheses of Theorem 1.
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