Colored HOMFLY polynomials via skein theory

Many knots and links in S 3 can be drawn as gluing of three manifolds with one or more four-punctured S 2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four -strand braids. Incorporating the properties of four-point conformal blocks in WZNW models, we conjecture colored HOMFLY polynomials for these double fat graphs where the color can be rectangular or non-rectangular representation. With the recent work of Gu-Jockers, the fusion matrices for the non-rectangular [21] representation, the first which involves multiplicity is known. We verify our conjecture by comparing with the [21] colored HOMFLY of many knots, obtained as closure of three braids. The conjectured form is computationally very effective leading to writing [21]-colored HOMFLY polynomials for many pretzel type knots and non-pretzel type knots. In particular, we find class of pretzel mutants which are distinguished and another class of mutants which cannot be distinguished by [21] representation. The difference between the [21]-colored HOMFLY of two mutants seems to have a general form, with A-dependence completely defined by the old conjecture due to Morton and Cromwell. In particular, we check it for an entire multi-parametric family of mutant knots evaluated using evolution method.

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“…As conjectured in [13,80] and proved in [81], the "special" polynomials, i.e. reduced HOM-FLY polynomials at q = 1, obey the factorization rule …”

Section: Special Polynomialsmentioning

confidence: 68%

Colored HOMFLY polynomials of knots presented as double fat diagrams

Миронов

Морозов

Ramadevi

et al. 2015

J. High Energ. Phys.

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“…This property (B.1) was proven for all knots in [87]. Recently, it was proposed that this could be lifted to the level of colored superpolynomials [62]:…”

Section: B Special Colored Superpolynomialsmentioning

confidence: 83%

Super-A-polynomials for twist knots

Nawata

Ramadevi

Zodinmawia

et al. 2012

J. High Energ. Phys.

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We conjecture formulae of the colored superpolynomials for a class of twist knots K p where p denotes the number of full twists. The validity of the formulae is checked by applying differentials and taking special limits. Using the formulae, we compute both the classical and quantum super-A-polynomials for the twist knots with small values of p. The results support the categorified versions of the generalized volume conjecture and the quantum volume conjecture. Furthermore, we obtain the evidence that the Q-deformed Apolynomials can be identified with the augmentation polynomials of knot contact homology in the case of the twist knots.

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“…We also do so for the figure eight knot 4 1 , where this provides a new set of colored HOMFLY polynomials and continuation to exceptional groups is a new result of its own value. Remarkably, the universal formulas inherit distinguished properties of ordinary knot polynomials like evolution [47,48], factorization of special polynomials [47,[49][50][51] and differential expansion [52][53][54].…”

Section: Universal Knot Polynomialsmentioning

confidence: 99%

On universal knot polynomials

Миронов

Mkrtchyan

Морозов

2016

J. High Energ. Phys.

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We present a universal knot polynomials for 2-and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel's plane, respectively and give their exceptional group's counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel's plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representations. Properties of universal polynomials and applications of these results are discussed.

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Colored HOMFLY polynomials via skein theory

Abstract: Abstract. In this paper, we study the properties of the colored HOMFLY polynomials via HOMFLY skein theory. We prove some limit behaviors and symmetries of the colored HOMFLY polynomial predicted in some physicists' recent works.

Cited by 36 publications

References 25 publications

Colored HOMFLY polynomials of knots presented as double fat diagrams

Colored HOMFLY polynomials of knots presented as double fat diagrams

Super-A-polynomials for twist knots

On universal knot polynomials

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