22 research outputs found
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
Racah coefficients and extended HOMFLY polynomials for all 5-, 6- and 7-strand braids
Basing on evaluation of the Racah coefficients for SU_q(3) (which supported
the earlier conjecture of their universal form) we derive explicit formulas for
all the 5-, 6- and 7-strand Wilson averages in the fundamental representation
of arbitrary SU(N) group (the HOMFLY polynomials). As an application, we list
the answers for all 5-strand knots with 9 crossings. In fact, the 7-strand
formulas are sufficient to reproduce all the HOMFLY polynomials from the
katlas.org: they are all described at once by a simple explicit formula with a
very transparent structure. Moreover, would the formulas for the relevant
SU_q(3) Racah coefficients remain true for all other quantum groups, the paper
provides a complete description of the fundamental HOMFLY polynomials for all
braids with any number of strands.Comment: 16 pages + Tables and Appendice
Matrix model version of AGT conjecture and generalized Selberg integrals
Operator product expansion (OPE) of two operators in two-dimensional
conformal field theory includes a sum over Virasoro descendants of other
operator with universal coefficients, dictated exclusively by properties of the
Virasoro algebra and independent of choice of the particular conformal model.
In the free field model, these coefficients arise only with a special
"conservation" relation imposed on the three dimensions of the operators
involved in OPE. We demonstrate that the coefficients for the three
unconstrained dimensions arise in the free field formalism when additional
Dotsenko-Fateev integrals are inserted between the positions of the two
original operators in the product. If such coefficients are combined to form an
-point conformal block on Riemann sphere, one reproduces the earlier
conjectured -ensemble representation of conformal blocks, thus proving
this (matrix model) version of the celebrated AGT relation. The statement can
also be regarded as a relation between the -symbols of the Virasoro algebra
and the slightly generalized Selberg integrals , associated with arbitrary
Young diagrams. The conformal blocks are multilinear combinations of such
integrals and the remaining part of the original AGT conjecture relates them to
the Nekrasov functions which have exactly the same structure.Comment: 19 page
HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
Explicit answer is given for the HOMFLY polynomial of the figure eight knot
in arbitrary symmetric representation R=[p]. It generalizes the old
answers for p=1 and 2 and the recently derived results for p=3,4, which are
fully consistent with the Ooguri-Vafa conjecture. The answer can be considered
as a quantization of the \sigma_R = \sigma_{[1]}^{|R|} identity for the
"special" polynomials (they define the leading asymptotics of HOMFLY at q=1),
and arises in a form, convenient for comparison with the representation of the
Jones polynomials as sums of dilogarithm ratios. In particular, we construct a
difference equation ("non-commutative A-polynomial") in the representation
variable p. Simple symmetry transformation provides also a formula for
arbitrary antisymmetric (fundamental) representation R=[1^p], which also passes
some obvious checks. Also straightforward is a deformation from HOMFLY to
superpolynomials. Further generalizations seem possible to arbitrary Young
diagrams R, but these expressions are harder to test because of the lack of
alternative results, even partial.Comment: 14 page
Colored HOMFLY Polynomials as Multiple Sums over Paths or Standard Young Tableaux
If a knot is represented by an m-strand braid, then HOMFLY polynomial in representation R is a sum over characters in all representations Q∈R⊗m. Coefficients in this sum are traces of products of quantum ℛ^-matrices along the braid, but these matrices act in the space of intertwiners, and their size is equal to the multiplicity MRQ of Q in R⊗m. If R is the fundamental representation R=[1]=□, then M□Q is equal to the number of paths in representation graph, which lead from the fundamental vertex □ to the vertex Q. In the basis of paths the entries of the m-1 relevant ℛ^-matrices are associated with the pairs of paths and are nonvanishing only when the two paths either coincide or differ by at most one vertex, as a corollary ℛ^-matrices consist of just 1×1 and 2×2 blocks, given by very simple explicit expressions. If cabling method is used to color the knot with the representation R, then the braid has as many as m|R| strands; Q have a bigger size m|R|, but only paths passing through the vertex R are included into the sums over paths which define the products and traces of the m|R|-1 relevant ℛ^-matrices. In the case of SU(N), this path sum formula can also be interpreted as a multiple sum over the standard Young tableaux. By now it provides the most effective way for evaluation of the colored HOMFLY polynomials, conventional or extended, for arbitrary braids