12,355 research outputs found
6j-symbols, hyperbolic structures and the Volume Conjecture
We compute the asymptotical growth rate of a large family of
-symbols and we interpret our results in geometric terms by relating them
to volumes of hyperbolic truncated tetrahedra. We address a question which is
strictly related with S.Gukov's generalized volume conjecture and deals with
the case of hyperbolic links in connected sums of . We answer
this question for the infinite family of fundamental shadow links.Comment: 17 pages, 3 figures. Published on Geometry & Topology 11 (2007
Categorified sl(N) invariants of colored rational tangles
We use categorical skew Howe duality to find recursion rules that compute
categorified sl(N) invariants of rational tangles colored by exterior powers of
the standard representation. Further, we offer a geometric interpretation of
these rules which suggests a connection to Floer theory. Along the way we make
progress towards two conjectures about the colored HOMFLY homology of rational
links.Comment: 45 pages, many figures, uses dcpic.sty, v2: minor changes and new
example 5
Asymptotics of classical spin networks
A spin network is a cubic ribbon graph labeled by representations of
. Spin networks are important in various areas of Mathematics
(3-dimensional Quantum Topology), Physics (Angular Momentum, Classical and
Quantum Gravity) and Chemistry (Atomic Spectroscopy). The evaluation of a spin
network is an integer number. The main results of our paper are: (a) an
existence theorem for the asymptotics of evaluations of arbitrary spin networks
(using the theory of -functions), (b) a rationality property of the
generating series of all evaluations with a fixed underlying graph (using the
combinatorics of the chromatic evaluation of a spin network), (c) rigorous
effective computations of our results for some -symbols using the
Wilf-Zeilberger theory, and (d) a complete analysis of the regular Cube
spin network (including a non-rigorous guess of its Stokes constants), in the
appendix.Comment: 24 pages, 32 figure
Skein theory and Witten-Reshetikhin-Turaev Invariants of links in lens spaces
We study the behavior of the Witten-Reshetikhin-Turaev SU(2) invariants of
links in L(p,q) as a function of the level r-2. They are given by 1 over the
square root of r times one of p Laurent polynomials evaluated at e to the 2 pi
i divided by 4pr. The congruence class of r modulo p determines which
polynomial is applicable. If p is zero modulo four, the meridian of L(p,q) is
non-trivial in the skein module but has trivial Witten-Reshetikhin-Turaev SU(2)
invariants. On the other hand, we show that one may recover the element in the
Kauffman bracket skein module of L(p,q) represented by a link from the
collection of the WRT invariants at all levels if p is a prime or twice an odd
prime. By a more delicate argument, this is also shown to be true for p=9.Comment: Much of the paper has been rewritten and simplified. The only if part
of theorem 2 is new. AMS-TeX, 10 page
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