Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed

Morozov, A.; Tselousov, N.

doi:10.48550/arxiv.2205.12238

Cited by 3 publications

(13 citation statements)

References 39 publications

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“…The result turns out to be modest if not discouraging -the values of a (δ) i appearing in these families look rather poor, at least for δ > 0. Still, the exercise is interesting and it also provides a more accurate formulation for the defect-preservation conjecture of [42] (see Section 6), which makes it fully consistent with the defect-degree correspondence studied in the present paper.…”

Section: Integralitysupporting

confidence: 79%

“…We restrict it to a symmetric representation R = [r] and refer to Section 3 of [42] for detailed notation in this case. We also do not put bars over n a , what is usually done to distinguish antiparallel evolution, since this is the only one which is matter for the formulas.…”

Section: Antiparallel Descendants Of Torus Knotsmentioning

confidence: 99%

“…More examples are given by the larger families of double braids [35] and 3-pretzels [42], also obtained by the defect-preserving antiparallel evolution from the trefoil 3 1 [53] and thus all having the defect zero:…”

Section: Defect δmentioning

confidence: 99%

“…The differential expansion (DE) for the reduced HOMFLY polynomial in symmetric representations for the figure-eight knot 4 1 was introduced in [34] and was further generalized to other knots [35][36][37][38][39][40][41][42]. Originally it was suggested in the form…”

Section: Introductionmentioning

confidence: 99%

“…The variables G K [j] can provide better coordinates in the space of knots than the HOMFLY polynomials themselves [50], still they are far from being free parameters. In particular, they satisfy the C-polynomial equations [42,51,52], which are still far from being well understood and classified.…”

Section: Introductionmentioning

confidence: 99%

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Defect and degree of the Alexander polynomial

Lanina¹,

Morozov²

2022

Preprint

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Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be alternatively described as the degree in q ±2 of the fundamental Alexander polynomial, which formally corresponds to the case of no colors. We also pose a question if these Alexander polynomials can be arbitrary integer polynomials of a given degree. A first attempt to answer the latter question is a preliminary analysis of antiparallel descendants of the 2-strand torus knots, which provide a nice set of examples for all values of the defect. The answer turns out to be positive in the case of defect zero knots, what can be observed already in the case of twist knots. This proved conjecture also allows us to provide a complete set of C-polynomials for the symmetrically colored Alexander polynomials for defect zero. In this case, we achieve a complete separation of representation and knot variables.

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Section: Integralitysupporting

confidence: 79%

Section: Antiparallel Descendants Of Torus Knotsmentioning

confidence: 99%

Section: Defect δmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Defect and degree of the Alexander polynomial

Lanina¹,

Morozov²

2022

Preprint

Self Cite

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Defect and degree of the Alexander polynomial

Lanina

Морозов

2022

Eur. Phys. J. C

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Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be alternatively described as the degree in $$q^{\pm 2}$$ q ± 2 of the fundamental Alexander polynomial, which formally corresponds to the case of no colors. We also pose a question if these Alexander polynomials can be arbitrary integer polynomials of a given degree. A first attempt to answer the latter question is a preliminary analysis of antiparallel descendants of the 2-strand torus knots, which provide a nice set of examples for all values of the defect. The answer turns out to be positive in the case of defect zero knots, what can be observed already in the case of twist knots. This proved conjecture also allows us to provide a complete set of C-polynomials for the symmetrically colored Alexander polynomials for defect zero. In this case, we achieve a complete separation of representation and knot variables.

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Supersymmetric partition function hierarchies and character expansions

Wang,

Liu,

et al. 2023

Eur. Phys. J. C

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We construct the supersymmetric $$\beta $$ β and (q, t)-deformed Hurwitz–Kontsevich partition functions through W-representations and present the corresponding character expansions with respect to the Jack and Macdonald superpolynomials, respectively. Based on the constructed $$\beta $$ β and (q, t)-deformed superoperators, we further give the supersymmetric $$\beta $$ β and (q, t)-deformed partition function hierarchies through W-representations. We also present the generalized super Virasoro constraints, where the constraint operators obey the generalized super Virasoro algebra and null super 3-algebra. Moreover, the superintegrability for these (non-deformed) supersymmetric hierarchies is shown by their character expansions, i.e., $$<character>\sim character$$ < c h a r a c t e r > ∼ c h a r a c t e r .

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Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed

Cited by 3 publications

References 39 publications

Defect and degree of the Alexander polynomial

Defect and degree of the Alexander polynomial

Defect and degree of the Alexander polynomial

Supersymmetric partition function hierarchies and character expansions

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