119174057
Link polynomial calculus and the AENV conjecture
A bstractUsing the recently proposed differential hierarchy (Z-expansion) technique, we obtain a general expression for the HOMFLY polynomials in two arbitrary symmetric representations of link families, including Whitehead and Borromean links. Among other things, this allows us to check and confirm the recent conjecture of [1] that the large representation limit (the same as considered in the knot volume conjecture) of this quantity matches the prediction from mirror symmetry consideration. We also provide, using the evolution method, the HOMFLY polynomial in two arbitrary symmetric representations for an arbitrary member of the one-parametric family of 2-component 3-strand links, which includes the Hopf and Whitehead links.
121203905
Check of AGT relation for conformal blocks on sphere
Abstract The AGT conjecture identifying conformal blocks with the Nekrasov functions is investigated for the spherical conformal blocks with more than 4 external legs. The diagram technique which arises in conformal block calculation involves propagators and vertices. We evaluated vertices with two Virasoro algebra descendants and explicitly checked the AGT relation up to the third order of the expansion for the 5-point and 6-point conformal blocks on sphere confirming all the predictions of arXiv:0906.3219 relevant in this situation. We propose that U ( 1 ) -factor can be extracted from the matrix elements of the free field vertex operators. We studied the n -point case, and found out that our results confirm the AGT conjecture up to the third order expansions.
119577066
HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
A bstractExplicit answer is given for the HOMFLY polynomial of the figure eight knot 41 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 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 $ \mathcal{A} $-polynomial”) in the representation variable p. Simple symmetry transformation provides also a formula for arbitrary antisymmetric (fundamental) representation R = [1p], 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.
116961882
Character expansion for HOMFLY polynomials. I. Integrability and difference equations
We suggest to associate with each knot the set of coefficients of its HOMFLY polynomial expansion into the Schur functions. For each braid representation of the knot these coefficients are defined unambiguously as certain combinations of the Racah symbols for the algebra SU_q. Then, the HOMFLY polynomials can be extended to the entire space of time-variables. The so extended HOMFLY polynomials are no longer knot invariants, they depend on the choice of the braid representation, but instead one can naturally discuss their explicit integrable properties. The generating functions of torus knot/link coefficients are turned to satisfy the Plucker relations and can be associated with tau-function of the KP hierarchy, while generic knots correspond to more involved systems. On the other hand, using the expansion into the Schur functions, one can immediately derive difference equations (A-polynomials) for knot polynomials which play a role of the string equation. This adds to the previously demonstrated use of these character decompositions for the study of beta-deformations from HOMFLY to superpolynomials.
119162645
Towards matrix model representation of HOMFLY polynomials
We investigate possibilities of generalizing the TBEM (Tierz, Brini-Eynard-Mariño) 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 Hermitian 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 ħ = logq. 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 τ-function approach to knot theory, but this remains a subject for future investigation.
119623409
On possible existence of HOMFLY polynomials for virtual knots
Abstract Virtual knots are associated with knot diagrams, which are not obligatory planar. The recently suggested generalization from N = 2 to arbitrary N of the Kauffman–Khovanov calculus of cycles in resolved diagrams can be straightforwardly applied to non-planar case. In simple examples we demonstrate that this construction preserves topological invariance – thus implying the existence of HOMFLY extension of cabled Jones polynomials for virtual knots and links.
15661590
The H1 forward proton spectrometer at HERA
Abstract The forward proton spectrometer is part of the H1 detector at the HERA collider. Protons with energies above 500 GeV and polar angles below 1 mrad can be detected by this spectrometer. The main detector components are scintillating fiber detectors read out by position-sensitive photo-multipliers. These detectors are housed in the so-called Roman Pots which allow them to be moved close to the circulating proton beam. Four Roman Pot stations are located at distances between 60 and 90 m from the interaction point.
119611129
Differential hierarchy and additional grading of knot polynomials
Colored knot polynomials have a special Z-expansion in certain combinations of differentials, which depend on the representation. The expansion coefficients are functions of three variables A, q, and t and can be regarded as new distinguished coordinates on the space of knot polynomials, analogous to the coefficients of the alternative character expansion. These new variables decompose especially simply when the representation is embedded into a product of fundamental representations. The recently proposed fourth grading is seemingly a simple redefinition of these new coordinates, elegant, but in no way distinguished. If this is so, then it does not provide any new independent knot invariants, but it can instead be regarded as one more piece of evidence in support of a hidden differential hierarchy (Z-expansion) structure behind the knot polynomials.