scholarly journals Khovanov homology detects the figure‐eight knot

2021 ◽  
Author(s):  
John A. Baldwin ◽  
Nathan Dowlin ◽  
Adam Simon Levine ◽  
Tye Lidman ◽  
Radmila Sazdanovic
Keyword(s):  
Khovanov Homology ◽  
Figure Eight Knot

scholarly journals Curved Rickard complexes and link homologies

2020 ◽  
Vol 2020 (769) ◽  
pp. 87-119
Author(s):  
Sabin Cautis ◽  
Aaron D. Lauda ◽  
Joshua Sussan
Keyword(s):  
Spectral Sequence ◽  
Quantum Groups ◽  
Braid Group ◽  
Derived Categories ◽  
Duke Math ◽  
Khovanov Homology ◽  
Hilbert Schemes ◽  
Coherent Sheaves ◽  
Knot Homology ◽  
Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

scholarly journals A Steenrod square on Khovanov homology

2014 ◽  
Vol 7 (3) ◽  
pp. 817-848 ◽  
Author(s):  
Robert Lipshitz ◽  
Sucharit Sarkar
Keyword(s):  
Khovanov Homology ◽  
Steenrod Square

scholarly journals On independence of iterated Whitehead doubles in the knot concordance group

2018 ◽  
Vol 27 (01) ◽  
pp. 1850003
Author(s):  
Kyungbae Park
Keyword(s):  
Correction Term ◽  
Khovanov Homology ◽  
Floer Theory ◽  
Knot Concordance ◽  
Torus Knot ◽  
Linearly Independent ◽  
Heegaard Floer

Let [Formula: see text] be the positively clasped untwisted Whitehead double of a knot [Formula: see text], and [Formula: see text] be the [Formula: see text] torus knot. We show that [Formula: see text] and [Formula: see text] are linearly independent in the smooth knot concordance group [Formula: see text] for each [Formula: see text]. Further, [Formula: see text] and [Formula: see text] generate a [Formula: see text] summand in the subgroup of [Formula: see text] generated by topologically slice knots. We use the concordance invariant [Formula: see text] of Manolescu and Owens, using Heegaard Floer correction term. Interestingly, these results are not easily shown using other concordance invariants such as the [Formula: see text]-invariant of knot Floer theory and the [Formula: see text]-invariant of Khovanov homology. We also determine the infinity version of the knot Floer complex of [Formula: see text] for any [Formula: see text] generalizing a result for [Formula: see text] of Hedden, Kim and Livingston.

scholarly journals HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations

2012 ◽  
Vol 2012 (7) ◽  
Author(s):  
H. Itoyama ◽  
A. Mironov ◽  
A. Morozov ◽  
And. Morozov
Keyword(s):  
Figure Eight Knot

scholarly journals Khovanov homology of torus links

2009 ◽  
Vol 156 (3) ◽  
pp. 533-541 ◽  
Author(s):  
Marko Stošić
Keyword(s):  
Khovanov Homology ◽  
Torus Links

scholarly journals Constructing 1-cusped isospectral non-isometric hyperbolic 3-manifolds

2017 ◽  
Vol 10 (01) ◽  
pp. 1-25
Author(s):  
Stavros Garoufalidis ◽  
Alan W. Reid
Keyword(s):  
Discrete Spectrum ◽  
Eisenstein Series ◽  
Finite Volume ◽  
Finite Covers ◽  
Figure Eight Knot ◽  
Complex Length

We construct infinitely many examples of pairs of isospectral but non-isometric [Formula: see text]-cusped hyperbolic [Formula: see text]-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada’s method in the cusped setting, and so in addition our pairs are finite covers of the same degree of a 1-cusped hyperbolic 3-orbifold (indeed manifold) and also have the same complex length spectra. Finally we prove that any finite volume hyperbolic 3-manifold isospectral to the figure-eight knot complement is homeomorphic to the figure-eight knot complement.

scholarly journals Volumes and Degeneration of Cone-structures on the Figure-eight Knot

2006 ◽  
Vol 29 (2) ◽  
pp. 445-464 ◽  
Author(s):  
Alexander MEDNYKH ◽  
Alexey RASSKAZOV
Keyword(s):  
Figure Eight Knot

scholarly journals Profinite Rigidity, Fibering, and the Figure-Eight Knot

What's Next? ◽  
2020 ◽  
pp. 45-64
Author(s):  
Martin R. Bridson ◽  
Alan W. Reid
Keyword(s):  
Figure Eight Knot
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