116 research outputs found
Topología algebraica basada en nudos
En este artículo se discuten varios problemas abiertos en teoría de nudos. Además se definen las p-coloraciones de Fox y se discuten los movimientos racionales en entrelazados para introducir la estructura simpléctica sobre la frontera de un entrelazado, de tal forma que los entrelazados den lugar a lagrangianos en el espacio simpléctico. El objetivo final es abordar los módulos de madeja, discutiendo la topología algebraica basada en los nudos e introduciendo los módulos de madeja relacionados con los 3-movimientos y la conjetura de Montesinos-Nakanishi. Se introducirá al lector a todos estos temas haciendo un pequeñoo recorrido en el mundo de los módulos de madeja
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
Complex maps without invariant densities
We consider complex polynomials for and
, and find some combinatorial types and values of such that
there is no invariant probability measure equivalent to conformal measure on
the Julia set. This holds for particular Fibonacci-like and Feigenbaum
combinatorial types when sufficiently large and also for a class of
`long-branched' maps of any critical order.Comment: Typos corrected, minor changes, principally to Section
Equilibrium states for potentials with \sup\phi - \inf\phi < \htop(f)
In the context of smooth interval maps, we study an inducing scheme approach
to prove existence and uniqueness of equilibrium states for potentials
with he `bounded range' condition \sup \phi - \inf \phi < \htop, first used
by Hofbauer and Keller. We compare our results to Hofbauer and Keller's use of
Perron-Frobenius operators. We demonstrate that this `bounded range' condition
on the potential is important even if the potential is H\"older continuous. We
also prove analyticity of the pressure in this context.Comment: Added Lemma 6 to deal with the disparity between leading eigenvalues
and operator norms. Added extra references and corrected some typo
Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
We continue the program of systematic study of extended HOMFLY polynomials.
Extended polynomials depend on infinitely many time variables, are close
relatives of integrable tau-functions, and depend on the choice of the braid
representation of the knot. They possess natural character decompositions, with
coefficients which can be defined by exhaustively general formula for any
particular number m of strands in the braid and any particular representation R
of the Lie algebra GL(\infty). Being restricted to "the topological locus" in
the space of time variables, the extended HOMFLY polynomials reproduce the
ordinary knot invariants. We derive such a general formula, for m=3, when the
braid is parameterized by a sequence of integers (a_1,b_1,a_2,b_2,...), and for
the first non-fundamental representation R=[2]. Instead of calculating the
mixing matrices directly, we deduce them from comparison with the known answers
for torus and composite knots. A simple reflection symmetry converts the answer
for the symmetric representation [2] into that for the antisymmetric one [1,1].
The result applies, in particular, to the figure eight knot 4_1, and was
further extended to superpolynomials in arbitrary symmetric and antisymmetric
representations in arXiv:1203.5978.Comment: 22 pages + Tables of knot polynomial
Inductive Construction of 2-Connected Graphs for Calculating the Virial Coefficients
In this paper we give a method for constructing systematically all simple
2-connected graphs with n vertices from the set of simple 2-connected graphs
with n-1 vertices, by means of two operations: subdivision of an edge and
addition of a vertex. The motivation of our study comes from the theory of
non-ideal gases and, more specifically, from the virial equation of state. It
is a known result of Statistical Mechanics that the coefficients in the virial
equation of state are sums over labelled 2-connected graphs. These graphs
correspond to clusters of particles. Thus, theoretically, the virial
coefficients of any order can be calculated by means of 2-connected graphs used
in the virial coefficient of the previous order. Our main result gives a method
for constructing inductively all simple 2-connected graphs, by induction on the
number of vertices. Moreover, the two operations we are using maintain the
correspondence between graphs and clusters of particles.Comment: 23 pages, 5 figures, 3 table
String theory and the Kauffman polynomial
We propose a new, precise integrality conjecture for the colored Kauffman
polynomial of knots and links inspired by large N dualities and the structure
of topological string theory on orientifolds. According to this conjecture, the
natural knot invariant in an unoriented theory involves both the colored
Kauffman polynomial and the colored HOMFLY polynomial for composite
representations, i.e. it involves the full HOMFLY skein of the annulus. The
conjecture sheds new light on the relationship between the Kauffman and the
HOMFLY polynomials, and it implies for example Rudolph's theorem. We provide
various non-trivial tests of the conjecture and we sketch the string theory
arguments that lead to it.Comment: 36 pages, many figures; references and examples added, typos
corrected, final version to appear in CM
On the dimension of graphs of Weierstrass-type functions with rapidly growing frequencies
We determine the Hausdorff and box dimension of the fractal graphs for a
general class of Weierstrass-type functions of the form , where is a periodic
Lipschitz real function and , as . Moreover, for any , we provide
examples of such functions with \dim_H(\graph f) = \underline{\dim}_B(\graph
f) = H, \bar{\dim}_B(\graph f) = B.Comment: 18 page
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