392 research outputs found
Finite Type Invariants of Classical and Virtual Knots
We observe that any knot invariant extends to virtual knots. The isotopy
classification problem for virtual knots is reduced to an algebraic problem
formulated in terms of an algebra of arrow diagrams. We introduce a new notion
of finite type invariant and show that the restriction of any such invariant of
degree n to classical knots is an invariant of degree at most n in the
classical sense. A universal invariant of degree at most n is defined via a
Gauss diagram formula. This machinery is used to obtain explicit formulas for
invariants of low degrees. The same technique is also used to prove that any
finite type invariant of classical knots is given by a Gauss diagram formula.
We introduce the notion of n-equivalence of Gauss diagrams and announce virtual
counter-parts of results concerning classical n-equivalence.Comment: 22 pages, many figure
Combinatorial Formulae for Vassiliev Invariants from Chern-Simons Gauge Theory
We analyse the perturbative series expansion of the vacuum expectation value
of a Wilson loop in Chern-Simons gauge theory in the temporal gauge. From the
analysis emerges the notion of the kernel of a Vassiliev invariant. The kernel
of a Vassiliev invariant of order n is not a knot invariant, since it depends
on the regular knot projection chosen, but it differs from a Vassiliev
invariant by terms that vanish on knots with n singular crossings. We
conjecture that Vassiliev invariants can be reconstructed from their kernels.
We present the general form of the kernel of a Vassiliev invariant and we
describe the reconstruction of the full primitive Vassiliev invariants at
orders two, three and four. At orders two and three we recover known
combinatorial expressions for these invariants. At order four we present new
combinatorial expressions for the two primitive Vassiliev invariants present at
this order.Comment: 73 pages, latex, epsf, 18 figures, 2 table
Diassociative algebras and Milnor's invariants for tangles
We extend Milnor's mu-invariants of link homotopy to ordered (classical or
virtual) tangles. Simple combinatorial formulas for mu-invariants are given in
terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves
corresponds to axioms of Loday's diassociative algebra. The relation of tangles
to diassociative algebras is formulated in terms of a morphism of corresponding
operads.Comment: 17 pages, many figures; v2: several typos correcte
Kontsevich integral for knots and Vassiliev invariants
We review quantum field theory approach to the knot theory. Using holomorphic
gauge we obtain the Kontsevich integral. It is explained how to calculate
Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial
way which can be programmed on a computer. We discuss experimental results and
temporal gauge considerations which lead to representation of Vassiliev
invariants in terms of arrow diagrams. Explicit examples and computational
results are presented.Comment: 25 pages, 17 figure
The Pure Virtual Braid Group Is Quadratic
If an augmented algebra K over Q is filtered by powers of its augmentation
ideal I, the associated graded algebra grK need not in general be quadratic:
although it is generated in degree 1, its relations may not be generated by
homogeneous relations of degree 2. In this paper we give a sufficient criterion
(called the PVH Criterion) for grK to be quadratic. When K is the group algebra
of a group G, quadraticity is known to be equivalent to the existence of a (not
necessarily homomorphic) universal finite type invariant for G. Thus the PVH
Criterion also implies the existence of such a universal finite type invariant
for the group G. We apply the PVH Criterion to the group algebra of the pure
virtual braid group (also known as the quasi-triangular group), and show that
the corresponding associated graded algebra is quadratic, and hence that these
groups have a (not necessarily homomorphic) universal finite type invariant.Comment: 53 pages, 15 figures. Some clarifications added and inaccuracies
corrected, reflecting suggestions made by the referee of the published
version of the pape
On 3d extensions of AGT relation
An extension of the AGT relation from two to three dimensions begins from
connecting the theory on domain wall between some two S-dual SYM models with
the 3d Chern-Simons theory. The simplest kind of such a relation would
presumably connect traces of the modular kernels in 2d conformal theory with
knot invariants. Indeed, the both quantities are very similar, especially if
represented as integrals of the products of quantum dilogarithm functions.
However, there are also various differences, especially in the "conservation
laws" for integration variables, which hold for the monodromy traces, but not
for the knot invariants. We also discuss another possibility: interpretation of
knot invariants as solutions to the Baxter equations for the relativistic Toda
system. This implies another AGT like relation: between 3d Chern-Simons theory
and the Nekrasov-Shatashvili limit of the 5d SYM.Comment: 23 page
A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres
We construct an invariant J_M of integral homology spheres M with values in a
completion \hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at
each root of unity \zeta gives the the SU(2) Witten-Reshetikhin-Turaev
invariant \tau_\zeta(M) of M at \zeta. Thus J_M unifies all the SU(2)
Witten-Reshetikhin-Turaev invariants of M. As a consequence, \tau_\zeta(M) is
an algebraic integer. Moreover, it follows that \tau_\zeta(M) as a function on
\zeta behaves like an ``analytic function'' defined on the set of roots of
unity. That is, the \tau_\zeta(M) for all roots of unity are determined by a
"Taylor expansion" at any root of unity, and also by the values at infinitely
many roots of unity of prime power orders. In particular, \tau_\zeta(M) for all
roots of unity are determined by the Ohtsuki series, which can be regarded as
the Taylor expansion at q=1.Comment: 66 pages, 8 figure
Modelling of the effect of ELMs on fuel retention at the bulk W divertor of JET
Effect of ELMs on fuel retention at the bulk W target of JET ITER-Like Wall was studied with multi-scale calculations. Plasma input parameters were taken from ELMy H-mode plasma experiment. The energetic intra-ELM fuel particles get implanted and create near-surface defects up to depths of few tens of nm, which act as the main fuel trapping sites during ELMs. Clustering of implantation-induced vacancies were found to take place. The incoming flux of inter-ELM plasma particles increases the different filling levels of trapped fuel in defects. The temperature increase of the W target during the pulse increases the fuel detrapping rate. The inter-ELM fuel particle flux refills the partially emptied trapping sites and fills new sites. This leads to a competing effect on the retention and release rates of the implanted particles. At high temperatures the main retention appeared in larger vacancy clusters due to increased clustering rate
On the mechanisms governing gas penetration into a tokamak plasma during a massive gas injection
A new 1D radial fluid code, IMAGINE, is used to simulate the penetration of gas into a tokamak plasma during a massive gas injection (MGI). The main result is that the gas is in general strongly braked as it reaches the plasma, due to mechanisms related to charge exchange and (to a smaller extent) recombination. As a result, only a fraction of the gas penetrates into the plasma. Also, a shock wave is created in the gas which propagates away from the plasma, braking and compressing the incoming gas. Simulation results are quantitatively consistent, at least in terms of orders of magnitude, with experimental data for a D 2 MGI into a JET Ohmic plasma. Simulations of MGI into the background plasma surrounding a runaway electron beam show that if the background electron density is too high, the gas may not penetrate, suggesting a possible explanation for the recent results of Reux et al in JET (2015 Nucl. Fusion 55 093013)
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