43 research outputs found
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
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
On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions
In the genus expansion of the HOMFLY polynomials their representation
dependence is naturally captured by symmetric group characters. This
immediately implies that the Ooguri-Vafa partition function (OVPF) is a Hurwitz
tau-function. In the planar limit involving factorizable special polynomials,
it is actually a trivial exponential tau-function. In fact, in the double
scaling Kashaev limit (the one associated with the volume conjecture) dominant
in the genus expansion are terms associated with the symmetric representations
and with the integrability preserving Casimir operators, though we stop one
step from converting this fact into a clear statement about the OVPF behavior
in the vicinity of q=1. Instead, we explain that the genus expansion provides a
hierarchical decomposition of the Hurwitz tau-function, similar to the
Takasaki-Takebe expansion of the KP tau-functions. This analogy can be helpful
to develop a substitute for the universal Grassmannian description in the
Hurwitz tau-functions.Comment: 8 page
Colored HOMFLY polynomials for the pretzel knots and links
With the help of the evolution method we calculate all HOMFLY polynomials in
all symmetric representations [r] for a huge family of (generalized) pretzel
links, which are made from g+1 two strand braids, parallel or antiparallel, and
depend on g+1 integer numbers. We demonstrate that they possess a pronounced
new structure: are decomposed into a sum of a product of g+1 elementary
polynomials, which are obtained from the evolution eigenvalues by rotation with
the help of rescaled SU_q(N) Racah matrix, for which we provide an explicit
expression. The generalized pretzel family contains many mutants,
undistinguishable by symmetric HOMFLY polynomials, hence, the extension of our
results to non-symmetric representations R is a challenging open problem. To
this end, a non-trivial generalization of the suggested formula can be
conjectured for entire family with arbitrary g and R.Comment: 26 pages + tables of pretzel knots up to 10 crossing