THE KHOVANOV HOMOLOGY OF (p, -p, q) PRETZEL KNOTS

We use the contact invariant defined in [2] to construct a new invariant of Legendrian knots in Kronheimer and Mrowka's monopole knot homology theory (KHM ), following a prescription of Stipsicz and Vértesi. Our Legendrian invariant improves upon an earlier Legendrian invariant in KHM defined by the second author in several important respects. Most notably, ours is preserved by negative stabilization. This fact enables us to define a transverse knot invariant in KHM via Legendrian approximation. It also makes our invariant a more likely candidate for the monopole Floer analogue of the "LOSS" invariant in knot Floer homology. Like its predecessor, our Legendrian invariant behaves functorially with respect to Lagrangian concordance. We show how this fact can be used to compute our invariant in several examples. As a byproduct of our investigations, we provide the first infinite family of nonreversible Lagrangian concordances between prime knots.

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“…Remark 5.5. We expect that L(K n ) = 0 for all n. Indeed, each K n has thin Khovanov homology [41]. Hence, by a well-known conjecture (cf.…”

Section: Examples and Nonreversible Lagrangian Concordancesmentioning

confidence: 91%

Invariants of Legendrian and transverse knots in monopole knot homology

Baldwin

Sivek

2018

Journal of Symplectic Geometry

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“…Starkston has conjectured [18] and Qazaqzeh [17] has shown that the Khovanov homology of P (−p, q, r) is thin if p = min{q, r}, and Manion [13] has proved that it is not thin if 2 ≤ p < min{q, r}.…”

Section: 2mentioning

confidence: 99%

On spectral sequences from Khovanov homology

Lobb

Zentner

2020

Algebr. Geom. Topol.

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There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphisms remain unexploited in the literature, perhaps because there is still an open question concerning the naturality of maps induced by general movies.In this paper we focus on the spectral sequences due to Kronheimer-Mrowka from Khovanov homology to instanton knot Floer homology, and on that due to Ozsváth-Szabó to the Heegaard-Floer homology of the branched double cover. For example, we use the 1-handle morphisms to give new information about the filtrations on the instanton knot Floer homology of the (4, 5)-torus knot, determining these up to an ambiguity in a pair of degrees; to determine the Ozsváth-Szabó spectral sequence for an infinite class of prime knots; and to show that higher differentials of both the Kronheimer-Mrowka and the Ozsváth-Szabó spectral sequences necessarily lower the delta grading for all pretzel knots.

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“…The next step was the explicit computation of unreduced Khovanov polynomials for several infinite series of non-quasi-alternating genus 2 pretzel links [34][35][36]. All these polynomials proved to be homologically thin, and thus similar to the polynomials of the alternating links.…”

Section: Khovanov Polynomials For Genus 2 Prezel Knotsmentioning

confidence: 99%

Nimble evolution for pretzel Khovanov polynomials

Anokhina¹,

Морозов

Popolitov

2019

Eur. Phys. J. C

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We conjecture explicit evolution formulas for Khovanov polynomials, which for any particular knot are Laurent polynomials of complex variables q and T , for pretzel knots of genus g in some regions in the space of winding parameters n 0 , . . . , ng.Our description is exhaustive for genera 1 and 2. As previously observed [14,15], evolution at T = −1 is not fully smooth: it switches abruptly at the boundaries between different regions. We reveal that this happens also at the boundary between thin and thick knots, moreover, the thick-knot domain is further stratified. For thin knots the two eigenvalues 1 and λ = q 2 T , governing the evolution, are the standard T -deformation of the eigenvalues of the R-matrix 1 and −q 2 . However, in thick knots' regions extra eigenvalues emerge, and they are powers of the "naive" λ, namely, they are equal to λ 2 , . . . , λ g . From point of view of frequencies, i.e. logarithms of eigenvalues, this is frequency doubling (more precisely, frequency multiplication) -a phenomenon typical for non-linear dynamics.Hence, our observation can signal a hidden non-linearity of superpolynomial evolution. To give this newly observed evolution a short name, note that when λ is pure phase the contributions of λ 2 , . . . , λ g oscillate "faster" than the one of λ. Hence, we call this type of evolution "nimble".

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THE KHOVANOV HOMOLOGY OF (p, -p, q) PRETZEL KNOTS

Cited by 8 publications

References 7 publications

Invariants of Legendrian and transverse knots in monopole knot homology

Invariants of Legendrian and transverse knots in monopole knot homology

On spectral sequences from Khovanov homology

Nimble evolution for pretzel Khovanov polynomials

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