On the formulae for the colored HOMFLY polynomials

Kawagoe, Kenichi

doi:10.1016/j.geomphys.2016.02.012

Cited by 13 publications

(18 citation statements)

References 11 publications

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“…Here the summations in (5.7) are indeed finite because F r,i (a, q) = 0 for i > r. We verified that the expressions (5.7) are consistent with the colored HOMFLY invariants of the pretzel knots [108], the Whitehead link [92,105] and the Borromean rings [92]. Furthermore, Habiro has introduced unified quantum invariants of 3-manifolds which become the WRT invariants at a root of unity [78].…”

Section: Cyclotomic Expansions Of Colored Homfly Invariantssupporting

confidence: 66%

“…Although the Rosso-Jones formulae [102][103][104] provide a closed form for quantum group invariants of torus links colored by arbitrary representations, it is very difficult to compute the colored quantum group invariants for other knots and links. So far, closed form expressions for the HOMFLY invariants colored by arbitrary symmetric representations have been obtained for only a small number of non-torus links [92,105]. As we show below, for a certain family of hyperbolic links, Habiro's cyclotomic expansions of colored Jones polynomials can be generalized not only to HOMFLY invariants colored by arbitrary symmetric representations, but also to Poincaré polynomials of putative link homologies.…”

Section: Cyclotomic Expansions Of Link Invariantsmentioning

confidence: 87%

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Sequencing BPS spectra

Gukov

Nawata

Saberi

et al. 2016

J. High Energ. Phys.

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This paper provides both a detailed study of color-dependence of link homologies, as realized in physics as certain spaces of BPS states, and a broad study of the behavior of BPS states in general. We consider how the spectrum of BPS states varies as continuous parameters of a theory are perturbed. This question can be posed in a wide variety of physical contexts, and we answer it by proposing that the relationship between unperturbed and perturbed BPS spectra is described by a spectral sequence. These general considerations unify previous applications of spectral sequence techniques to physics, and explain from a physical standpoint the appearance of many spectral sequences relating various link homology theories to one another. We also study structural properties of colored HOMFLY homology for links and evaluate Poincaré polynomials in numerous examples. Among these structural properties is a novel "sliding" property, which can be explained by using (refined) modular S-matrix. This leads to the identification of modular transformations in Chern-Simons theory and 3d N = 2 theory via the 3d/3d correspondence. Lastly, we introduce the notion of associated varieties as classical limits of recursion relations of colored superpolynomials of links, and study their properties.

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Section: Cyclotomic Expansions Of Colored Homfly Invariantssupporting

confidence: 66%

Section: Cyclotomic Expansions Of Link Invariantsmentioning

confidence: 87%

Sequencing BPS spectra

Gukov

Nawata

Saberi

et al. 2016

J. High Energ. Phys.

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“…For example, to test the assertions of our recent work [19], it would be interesting to have HOMFLY polynomials for hyperbolic knots colored with Young diagrams with up to two rows at our disposal. There are some results on HOMFLY polynomials for certain hyperbolic knots colored with totally symmetric and/or anti-symmetric representations [20][21][22][23][24][25][26]. More recently, for certain classes of knots HOMFLY invariants for colorings with more general representations have explicitly been obtained in refs.…”

Section: Introductionmentioning

confidence: 99%

A Note on Colored HOMFLY Polynomials for Hyperbolic Knots from WZW Models

Gu¹,

Jockers²

2015

Commun. Math. Phys.

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Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY invariants in terms of crossing matrices of the underlying Wess-Zumino-Witten model. Our analysis extends previous works by incorporating non-trivial multiplicities for the primaries appearing in the crossing matrices, so as to describe colorings of HOMFLY invariants beyond the totally symmetric or anti-symmetric representations of SU(N). The crossing matrices directly relate to 6j-symbols of the quantum group U q su(N). We present powerful methods to calculate such quantum 6j-symbols for general N. This allows us to determine previously unknown colored HOMFLY polynomials for two-bridge hyperbolic knots. We give explicitly the HOM-FLY polynomials colored by the representation {2, 1} for two-bridge hyperbolic knots with up to eight crossings. Yet, the scope of application of our techniques goes beyond knot theory; e.g., our findings can be used to study correlators in Wess-Zumino-Witten conformal field theories or -in the limit to classical groups -to determine color factors for Yang Mills amplitudes. July, 20142 The choice of k 1 and n is slightly different from ref.[21] to ensure that the charge n is integral. The regularization of the unnormalized colored HOMFLY invariants is nonetheless the same.

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“…Computations and questions. With regards to computation of the 4-variable polynomial of a knot, there are several formulas for the HOMFLYPT polynomial of some links in the literature colored by partitions with one row, see for example [Kaw,NRZS12,IMMM12b,IMMM12a]. These formulas are manifestly q-holonomic, as follows by the fundamental theorem of WZ theory.…”

Section: 5mentioning

confidence: 99%

The colored HOMFLYPT function is q-holonomic

Garoufalidis¹,

Lauda²,

Le³

2018

Duke Math. J.

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We prove that the HOMFLYPT polynomial of a link, colored by partitions with a fixed number of rows is a q-holonomic function. Specializing to the case of knots colored by a partition with a single row, it proves the existence of an (a, q) super-polynomial of knots in 3-space, as was conjectured by string theorists. Our proof uses skew Howe duality that reduces the evaluation of web diagrams and their ladders to a Poincare-Birkhoff-Witt computation of an auxiliary quantum group of rank the number of strings of the ladder diagram. The result is a concrete and algorithmic web evaluation algorithm that is manifestly q-holonomic.

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On the formulae for the colored HOMFLY polynomials

Cited by 13 publications

References 11 publications

Sequencing BPS spectra

Sequencing BPS spectra

A Note on Colored HOMFLY Polynomials for Hyperbolic Knots from WZW Models

The colored HOMFLYPT function is q-holonomic

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