620 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
Towards topological quantum computer
One of the principal obstacles on the way to quantum computers is the lack of
distinguished basis in the space of unitary evolutions and thus the lack of the
commonly accepted set of basic operations (universal gates). A natural choice,
however, is at hand: it is provided by the quantum R-matrices, the entangling
deformations of non-entangling (classical) permutations, distinguished from the
points of view of group theory, integrable systems and modern theory of
non-perturbative calculations in quantum field and string theory. Observables
in this case are (square modules of) the knot polynomials, and their pronounced
integrality properties could provide a key to error correction. We suggest to
use R-matrices acting in the space of irreducible representations, which are
unitary for the real-valued couplings in Chern-Simons theory, to build a
topological version of quantum computing.Comment: 14 page
Covariance of WDVV equations
The (generalized) WDVV equations for the prepotentials in topological
and Seiberg-Witten models are covariant with respect to non-linear
transformations, described in terms of solutions of associated linear problem.
Both time-variables and the prepotential change non-trivially, but period
matrix (prepotential's second derivatives) remains intact.Comment: LaTeX, 7 pages, no figures (misprints corrected
Towards effective topological field theory for knots
Construction of (colored) knot polynomials for double-fat graphs is further
generalized to the case when "fingers" and "propagators" are substituting
R-matrices in arbitrary closed braids with m-strands. Original version of
arXiv:1504.00371 corresponds to the case m=2, and our generalizations sheds
additional light on the structure of those mysterious formulas. Explicit
expressions are now combined from Racah matrices of the type and mixing matrices in the sectors
. Further extension is provided by composition
rules, allowing to glue two blocks, connected by an m-strand braid (they
generalize the product formula for ordinary composite knots with m=1).Comment: 10 pages + table in Appendi
Correlators in tensor models from character calculus
We explain how the calculations of arXiv:1704.08648, which provided the first
evidence for non-trivial structures of Gaussian correlators in tensor models,
are efficiently performed with the help of the (Hurwitz) character calculus.
This emphasizes a close similarity between technical methods in matrix and
tensor models and supports a hope to understand the emerging structures in very
similar terms. We claim that the -fold Gaussian correlators of rank
tensors are given by -linear combinations of dimensions with the Young
diagrams of size . The coefficients are made from the characters of the
symmetric group and their exact form depends on the symmetries of the
model. As the simplest application of this new knowledge, we provide simple
expressions for correlators in the Aristotelian tensor model as tri-linear
combinations of dimensions.Comment: 9 page
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