14,864 research outputs found
A volume-ish theorem for the Jones polynomial of alternating knots
The Volume conjecture claims that the hyperbolic Volume of a knot is
determined by the colored Jones polynomial.
The purpose of this article is to show a Volume-ish theorem for alternating
knots in terms of the Jones polynomial, rather than the colored Jones
polynomial: The ratio of the Volume and certain sums of coefficients of the
Jones polynomial is bounded from above and from below by constants.
Furthermore, we give experimental data on the relation of the growths of the
hyperbolic volume and the coefficients of the Jones polynomial, both for
alternating and non-alternating knots.Comment: 14 page
Oriented paths in n-chromatic digraphs
In this thesis, we try to treat the problem of oriented paths in n-chromatic
digraphs. We first treat the case of antidirected paths in 5-chromatic
digraphs, where we explain El-Sahili's theorem and provide an elementary and
shorter proof of it. We then treat the case of paths with two blocks in
n-chromatic digraphs with n greater than 4, where we explain the two different
approaches of Addario-Berry et al. and of El-Sahili. We indicate a mistake in
Addario-Berry et al.'s proof and provide a correction for it.Comment: 25 pages, Master thesis in Graph Theory at the Lebanese Universit
Just Renormalizable TGFT's on U(1)^d with Gauge Invariance
We study the polynomial Abelian or U(1)^d Tensorial Group Field Theories
equipped with a gauge invariance condition in any dimension d. From our
analysis, we prove the just renormalizability at all orders of perturbation of
the phi^4_6 and phi^6_5 random tensor models. We also deduce that the phi^4_5
tensor model is super-renormalizable.Comment: 33 pages, 22 figures. One added paragraph on the different notions of
connectedness, preciser formulation of the proof of the power counting
theorem, more general statements about traciality of tensor graph
Enhancing non-melonic triangulations: A tensor model mixing melonic and planar maps
Ordinary tensor models of rank are dominated at large by
tree-like graphs, known as melonic triangulations. We here show that
non-melonic contributions can be enhanced consistently, leading to different
types of large limits. We first study the most generic quartic model at
, with maximally enhanced non-melonic interactions. The existence of the
expansion is proved and we further characterize the dominant
triangulations. This combinatorial analysis is then used to define a
non-quartic, non-melonic class of models for which the large free energy
and the relevant expectations can be calculated explicitly. They are matched
with random matrix models which contain multi-trace invariants in their
potentials: they possess a branched polymer phase and a 2D quantum gravity
phase, and a transition between them whose entropy exponent is positive.
Finally, a non-perturbative analysis of the generic quartic model is performed,
which proves analyticity in the coupling constants in cardioid domains
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