Colored knot polynomials for arbitrary pretzel knots and links

Galakhov, D.; Melnikov, D.; Mironov, A.; Morozov, A.; Sleptsov, A.

doi:10.1016/j.physletb.2015.02.029

Title: Colored knot polynomials for arbitrary pretzel knots and links

Full Record
Other Related Research

Abstract

A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich (g+1)-parametric family of pretzel knots and links. The answer for the Jones and HOMFLY is fully and explicitly expressed through the Racah matrix of Uq(SU_N), and looks related to a modular transformation of toric conformal block. Knot polynomials are among the hottest topics in modern theory. They are supposed to summarize nicely representation theory of quantum algebras and modular properties of conformal blocks. The result reported in the present letter, provides a spectacular illustration and support to this general expectation.

Authors:: Galakhov, D.; Melnikov, D.; Mironov, A.; Morozov, A.; Sleptsov, A.

Publication Date:: Wed Apr 01 00:00:00 EDT 2015

Research Org.:: Rutgers Univ., Piscataway, NJ (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1198587

Alternate Identifier(s):: OSTI ID: 1188772

Grant/Contract Number:: SC0010008; ARRA-SC0003883; SC0007897

Resource Type:: Published Article

Journal Name:: Physics Letters B

Additional Journal Information:: Journal Name: Physics Letters B Journal Volume: 743 Journal Issue: C; Journal ID: ISSN 0370-2693

Publisher:: Elsevier

Country of Publication:: Netherlands

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING

Citation Formats


                    Galakhov, D., Melnikov, D., Mironov, A., Morozov, A., and Sleptsov, A. Colored knot polynomials for arbitrary pretzel knots and links.  Netherlands: N. p., 2015. 
Web.  doi:10.1016/j.physletb.2015.02.029.

Copy to clipboard


                    Galakhov, D., Melnikov, D., Mironov, A., Morozov, A., & Sleptsov, A. Colored knot polynomials for arbitrary pretzel knots and links.  Netherlands.  https://doi.org/10.1016/j.physletb.2015.02.029

Copy to clipboard


                    Galakhov, D., Melnikov, D., Mironov, A., Morozov, A., and Sleptsov, A. Wed .  
"Colored knot polynomials for arbitrary pretzel knots and links".  Netherlands.  https://doi.org/10.1016/j.physletb.2015.02.029.

Copy to clipboard


                    
@article{osti_1198587,

  title        = {Colored knot polynomials for arbitrary pretzel knots and links},

  author       = {Galakhov, D. and Melnikov, D. and Mironov, A. and Morozov, A. and Sleptsov, A.},

  abstractNote = {A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich (g+1)-parametric family of pretzel knots and links. The answer for the Jones and HOMFLY is fully and explicitly expressed through the Racah matrix of Uq(SUN), and looks related to a modular transformation of toric conformal block. Knot polynomials are among the hottest topics in modern theory. They are supposed to summarize nicely representation theory of quantum algebras and modular properties of conformal blocks. The result reported in the present letter, provides a spectacular illustration and support to this general expectation.},

  doi          = {10.1016/j.physletb.2015.02.029},

  journal      = {Physics Letters B},

  number       = C,

  volume       = 743,

  place        = {Netherlands},

  year         = {Wed Apr 01 00:00:00 EDT 2015},

  month        = {Wed Apr 01 00:00:00 EDT 2015}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Publisher's Version of Record
https://doi.org/10.1016/j.physletb.2015.02.029

Other availability

Search WorldCat to find libraries that may hold this journal

Citation Metrics:

Cited by: 39 works

Citation information provided by
Web of Science

Save / Share:

Export Metadata

Save to My Library

Similar Records in DOE PAGES and OSTI.GOV collections:

Knot invariants from Virasoro related representation and pretzel knots

Journal Article Galakhov, D. ; Melnikov, D. ; Mironov, A. ; ... - Nuclear Physics. B

In this study, we remind the method to calculate colored Jones polynomials for the plat representations of knot diagrams from the knowledge of modular transformation (monodromies) of Virasoro conformal blocks with insertions of degenerate fields. As an illustration we use a rich family of pretzel knots, lying on a surface of arbitrary genus g, which was recently analyzed by the evolution method. Further generalizations can be to generic Virasoro modular transformations, provided by integral kernels, which can lead to the Hikami invariants.
Cited by 26
https://doi.org/10.1016/j.nuclphysb.2015.07.035
Wall-crossing invariants: from quantum mechanics to knots

Journal Article Galakhov, D., E-mail: galakhov@itep.ru, E-mail: galakhov@physics.rutgers.edu ; Mironov, A., E-mail: mironov@lpi.ru ; Morozov, A., E-mail: morozov@itep.ru - Journal of Experimental and Theoretical Physics

We offer a pedestrian-level review of the wall-crossing invariants. The story begins from the scattering theory in quantum mechanics where the spectrum reshuffling can be related to permutations of S-matrices. In nontrivial situations, starting from spin chains and matrix models, the S-matrices are operatorvalued and their algebra is described in terms of R- and mixing (Racah) U-matrices. Then the Kontsevich-Soibelman (KS) invariants are nothing but the standard knot invariants made out of these data within the Reshetikhin-Turaev-Witten approach. The R and Racah matrices acquire a relatively universal form in the semiclassical limit, where the basic reshufflings with the change ofmore »« less
https://doi.org/10.1134/S1063776115030206
Kauffman knot polynomials in classical abelian Chern-Simons field theory

Journal Article Liu Xin, E-mail: liuxin@maths.usyd.edu.a - Annals of Physics (New York)

Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant t{sup I(L)} is constructed for a link L, where I is the abelian Chern-Simons action and t a formal constant. For oriented knotted vortex lines, t{sup I} satisfies the skein relations of the Kauffman R-polynomial; for un-oriented knotted lines, t{sup I} satisfies the skein relations of the Kauffman bracket polynomial. As an example the bracket polynomials of trefoil knots are computed, and the Jones polynomial is constructed from the bracket polynomial.
The properties of conformal blocks, the AGT hypothesis, and knot polynomials

Journal Article Morozov, A. A., E-mail: andrey.morozov@itep.ru ; National Research Nuclear University MEPhI - Physics of Particles and Nuclei

Various properties of correlators of the two-dimensional conformal field theory are discussed. Specifically, their relation to the partition function of the four-dimensional supersymmetric theory is analyzed. In addition to being of interest in its own right, this relation is of practical importance. For example, it is much easier to calculate the known expressions for the partition function of supersymmetric theory than to calculate directly the expressions for correlators in conformal theory. The examined representation of conformal theory correlators as a matrix model serves the same purpose. The integral form of these correlators allows one to generalize the obtained results formore »« less
https://doi.org/10.1134/S106377961605004X
Poincaré series, 3D gravity and CFT spectroscopy

Journal Article Keller, Christoph A. ; Maloney, Alexander - Journal of High Energy Physics (Online)

Modular invariance strongly constrains the spectrum of states of two dimensional conformal field theories. By summing over the images of the modular group, we construct candidate CFT partition functions that are modular invariant and have positive spectrum. This allows us to efficiently extract the constraints on the CFT spectrum imposed by modular invariance, giving information on the spectrum that goes beyond the Cardy growth of the asymptotic density of states. Some of the candidate modular invariant partition functions we construct have gaps of size (c - 1)/12, proving that gaps of this size and smaller are consistent with modular invariance.more »« less
Cited by 72
https://doi.org/10.1007/JHEP02(2015)080

Full Text Available

Similar Records