scholarly journals ON THE QUANTUM FILTRATION OF THE UNIVERSAL sl(2) FOAM COHOMOLOGY

2012 ◽  
Vol 21 (03) ◽  
pp. 1250023 ◽  
Author(s):  
CARMEN CAPRAU
Keyword(s):  
Lower Bound ◽  
Spectral Sequence ◽  
Khovanov Homology ◽  
Reidemeister Moves

We investigate the filtered theory corresponding to the universal sl(2) foam cohomology [Formula: see text] for links, where a, h ∈ ℂ. We show that there is a spectral sequence converging to [Formula: see text] which is invariant under the Reidemeister moves, and whose E1 term is isomorphic to Khovanov homology. This spectral sequence can be used to obtain from the foam perspective an analogue of the Rasmussen invariant and a lower bound for the slice genus of a 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 VOLUME FORM ON THE KHOVANOV INVARIANT

2012 ◽  
Vol 21 (04) ◽  
pp. 1250032
Author(s):  
JUAN ORTIZ-NAVARRO
Keyword(s):  
Chain Complex ◽  
Khovanov Homology ◽  
Volume Form ◽  
Reidemeister Torsion ◽  
Homology Groups ◽  
Numerical Invariant ◽  
Reidemeister Moves ◽  
Invariant Volume ◽  
Knots And Links

The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over ℤ is recovered.

scholarly journals A link-splitting spectral sequence in Khovanov homology

2015 ◽  
Vol 164 (5) ◽  
pp. 801-841 ◽  
Author(s):  
Joshua Batson ◽  
Cotton Seed
Keyword(s):  
Spectral Sequence ◽  
Khovanov Homology

Reidemeister moves and a polynomial of virtual knot diagrams

2015 ◽  
Vol 24 (02) ◽  
pp. 1550010 ◽  
Author(s):  
Myeong-Ju Jeong
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Virtual Knot ◽  
Reidemeister Moves ◽  
Knot Diagrams

In 2006 C. Hayashi gave a lower bound for the number of Reidemeister moves in deformation of two equivalent knot diagrams by using writhe and cowrithe. It can be naturally extended for two virtually isotopic virtual knot diagrams. We introduce a polynomial qK(t) of a virtual knot diagram K and give lower bounds for the number of Reidemeister moves in deformation of two virtually isotopic knots by using qK(t). We give an example which shows that the polynomial qK(t) is useful to map out a sequence of Reidemeister moves to deform a virtual knot diagram to another virtually isotopic one.

CHAIN HOMOTOPY MAPS FOR KHOVANOV HOMOLOGY

2011 ◽  
Vol 20 (01) ◽  
pp. 127-139
Author(s):  
NOBORU ITO
Keyword(s):  
Khovanov Homology ◽  
Reidemeister Moves ◽  
Chain Homotopy ◽  
Chain Maps

Explicit chain homotopy maps and chain maps for the Reidemeister moves of Khovanov homology are often useful for several proofs of the isotopy invariance of Khovanov homology. However, such maps are missing except for the first Reidemeister moves given by Viro. In this paper, such chain homotopy maps and chain maps are obtained explicitly for the second and third Reidemeister moves (Sec. 2). Some applications are given to show the usefulness of these maps (Sec. 3).

Linking invariants of even virtual links

2017 ◽  
Vol 26 (12) ◽  
pp. 1750072 ◽  
Author(s):  
Haruko A. Miyazawa ◽  
Kodai Wada ◽  
Akira Yasuhara
Keyword(s):  
Lower Bound ◽  
The Other ◽  
Virtual Link ◽  
Link Diagram ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Classical Link ◽  
Virtual Links ◽  
The Difference

A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

scholarly journals Khovanov homology and knot Floer homology for pointed links

2017 ◽  
Vol 26 (02) ◽  
pp. 1740004 ◽  
Author(s):  
John A. Baldwin ◽  
Adam Simon Levine ◽  
Sucharit Sarkar
Keyword(s):  
Spectral Sequence ◽  
Floer Homology ◽  
Khovanov Homology ◽  
Knot Floer Homology ◽  
Text Page ◽  
Text Component

A well-known conjecture states that for any [Formula: see text]-component link [Formula: see text] in [Formula: see text], the rank of the knot Floer homology of [Formula: see text] (over any field) is less than or equal to [Formula: see text] times the rank of the reduced Khovanov homology of [Formula: see text]. In this paper, we describe a framework that might be used to prove this conjecture. We construct a modified version of Khovanov homology for links with multiple basepoints and show that it mimics the behavior of knot Floer homology. We also introduce a new spectral sequence converging to knot Floer homology whose [Formula: see text] page is conjecturally isomorphic to our new version of Khovanov homology; this would prove that the conjecture stated above holds over the field [Formula: see text].

scholarly journals A lower bound for the number of Reidemeister moves of type III

2006 ◽  
Vol 153 (15) ◽  
pp. 2788-2794 ◽  
Author(s):  
J. Scott Carter ◽  
Mohamed Elhamdadi ◽  
Masahico Saito ◽  
Shin Satoh
Keyword(s):  
Lower Bound ◽  
Type Iii ◽  
Reidemeister Moves
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