9,967 research outputs found
Evolution method and "differential hierarchy" of colored knot polynomials
We consider braids with repeating patterns inside arbitrary knots which
provides a multi-parametric family of knots, depending on the "evolution"
parameter, which controls the number of repetitions. The dependence of knot
(super)polynomials on such evolution parameters is very easy to find. We apply
this evolution method to study of the families of knots and links which include
the cases with just two parallel and anti-parallel strands in the braid, like
the ordinary twist and 2-strand torus knots/links and counter-oriented 2-strand
links. When the answers were available before, they are immediately reproduced,
and an essentially new example is added of the "double braid", which is a
combination of parallel and anti-parallel 2-strand braids. This study helps us
to reveal with the full clarity and partly investigate a mysterious
hierarchical structure of the colored HOMFLY polynomials, at least, in
(anti)symmetric representations, which extends the original observation for the
figure-eight knot to many (presumably all) knots. We demonstrate that this
structure is typically respected by the t-deformation to the superpolynomials.Comment: 31 page
Gaussian distribution of LMOV numbers
Recent advances in knot polynomial calculus allowed us to obtain a huge
variety of LMOV integers counting degeneracy of the BPS spectrum of topological
theories on the resolved conifold and appearing in the genus expansion of the
plethystic logarithm of the Ooguri-Vafa partition functions. Already the very
first look at this data reveals that the LMOV numbers are randomly distributed
in genus (!) and are very well parameterized by just three parameters depending
on the representation, an integer and the knot. We present an accurate
formulation and evidence in support of this new puzzling observation about the
old puzzling quantities. It probably implies that the BPS states, counted by
the LMOV numbers can actually be composites made from some still more
elementary objects.Comment: 23 page
Racah matrices and hidden integrability in evolution of knots
We construct a general procedure to extract the exclusive Racah matrices S
and \bar S from the inclusive 3-strand mixing matrices by the evolution method
and apply it to the first simple representations R =[1], [2], [3] and [2,2].
The matrices S and \bar S relate respectively the maps (R\otimes R)\otimes \bar
R\longrightarrow R with R\otimes (R \otimes \bar R) \longrightarrow R and
(R\otimes \bar R) \otimes R \longrightarrow R with R\otimes (\bar R \otimes R)
\longrightarrow R. They are building blocks for the colored HOMFLY polynomials
of arbitrary arborescent (double fat) knots. Remarkably, the calculation
realizes an unexpected integrability property underlying the evolution
matrices.Comment: 16 page
Towards matrix model representation of HOMFLY polynomials
We investigate possibilities of generalizing the TBEM eigenvalue matrix
model, which represents the non-normalized colored HOMFLY polynomials for torus
knots as averages of the corresponding characters. We look for a model of the
same type, which is a usual Chern-Simons mixture of the Gaussian potential,
typical for Hermitean models, and the sine Vandermonde factors, typical for the
unitary ones. We mostly concentrate on the family of twist knots, which
contains a single torus knot, the trefoil. It turns out that for the trefoil
the TBEM measure is provided by an action of Laplace exponential on the Jones
polynomial. This procedure can be applied to arbitrary knots and provides a
TBEM-like integral representation for the N=2 case. However, beyond the torus
family, both the measure and its lifting to larger N contain non-trivial
corrections in \hbar=\log q. A possibility could be to absorb these corrections
into a deformation of the Laplace evolution by higher Casimir and/or
cut-and-join operators, in the spirit of Hurwitz tau-function approach to knot
theory, but this remains a subject for future investigation.Comment: 10 page
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