Evolution properties of the knot’s defect

Морозов, А.; Tselousov, N.

doi:10.1140/epjc/s10052-022-10705-2

Cited by 4 publications

(6 citation statements)

References 53 publications

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“…Still, triple pretzels provide additional curious examples. As already mentioned in [53], there are knots with the unit Alexander polynomial, a 0 = 0, in this family, which are not unknots, for example 6…”

Section: Descendants Of 3mentioning

confidence: 64%

“…2) provides just one point per defect in m = δ + 1-dimensional space of parameters a m−1 i . However, this may be not a big problem: we can now use the defect-preserving antiparallel evolution [53] in each intersection, which gives rise to a 2m + 1-dimensional family of antiparallel pretzels with the same defect δ = m − 1. This dimension is more than enough, the question is only if arbitrary integer vectors a m−1 i appear in this family.…”

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%

“…What still can be free are the fundamental coefficients G [1] (1, q) = F [1] (1, q) 1 at r = 1 -they could be arbitrary integer symmetric polynomials of degree δ. We show that this is indeed the case for δ = 0 and provide some ideas on how this problem can be investigated for higher δ. Namely, we use the defect-preserving antiparallel evolution [53] of the 2-strand knots, which provides rich sets of examples for all values of the defect. This also allows us to highlight the properties of the antiparallel evolution and derive non-trivial examples, when the defect drops down at particular points (the conjecture of [53] does not allow it to raise but allows occasional dropdowns).…”

Section: Introductionmentioning

confidence: 99%

“…• Second, is the defect indeed as simple as described in Section 3, or is there a more sophisticated structure, associated with more complicated degeneracy diagrams? We do not go deep into this problem (raised already in [49] and [53]), we just provide some illustrations of "accidental" degeneracies in Section 6. This also helps to reformulate the statement about invariance of the defect under antiparallel evolution more carefully -defect can actually drop down at some evolution points, and there remains a question, if this should be treated as "accidents" or as a manifestation of additional structures, promoting defect from just a number to a slightly more involved characteristic of the DE.…”

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: Descendants Of 3mentioning

confidence: 64%

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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Toward a theory of Yangians

Galakhov,

Morozov,

Tselousov

2024

Phys. Rev. D

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We review the main ideas underlying the emerging theory of Yangians—the new type of hidden symmetry in string-inspired models. Their classification by quivers is a far-going generalization of simple Lie algebras classification by Dynkin diagrams. However, this is still a kind of project, while a more constructive approach goes through toric Calabi-Yau spaces, related supersymmetric systems, and the Duistermaat-Heckman or equivariant integrals between the fixed points in the Atiyah-Drinfeld-Hitchin-Manin (ADHM)-like moduli spaces. These fixed points are classified by crystals (Young-type diagrams), and Yangian generators describe “instanton” transitions between them. Detailed examples will be presented elsewhere. Published by the American Physical Society 2024

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

Морозов

Tselousov

2022

Eur. Phys. J. C

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For a peculiar family of double braid knots there is a remarkable factorization formula for the coefficients of the differential (cyclotomic) expansion (DE), which nowadays is widely used to construct the exclusive Racah matrices S and $${\bar{S}}$$ S ¯ in arbitrary representations. The origins of the factorization remain obscure and the special role of double braids remains a mystery. In an attempt to broaden the perspective, we extend the family of double braids to antiparallel triple pretzels, which are obtained by the defect-preserving deformation from the trefoil and all have defect zero. It turns out that factorization of DE coefficients is violated quite strongly, still remains described by an elegant formula, at least for all symmetric representations.

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Evolution properties of the knot’s defect

Cited by 4 publications

References 53 publications

Defect and degree of the Alexander polynomial

Defect and degree of the Alexander polynomial

Toward a theory of Yangians

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

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