scholarly journals 2-Knot homology and Yoshikawa move

2020 ◽  
Vol 29 (09) ◽  
pp. 2050067
Author(s):  
Hiroshi Matsuda
Keyword(s):  
Contact Homology ◽  
Knot Homology ◽  
Marked Graph

Ng constructed an invariant of knots in [Formula: see text], a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in [Formula: see text] using marked graph diagrams.

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).

CYLINDRICAL CONTACT HOMOLOGY OF A DEHN TWIST

2009 ◽  
Vol 20 (12) ◽  
pp. 1479-1525
Author(s):  
MEI-LIN YAU
Keyword(s):  
Dehn Twist ◽  
Open Book ◽  
Contact Homology

We use open book representations of contact 3-manifolds to compute the cylindrical contact homology of a Stein-fillable contact 3-manifold represented by the open book whose monodromy is a positive Dehn twist on a torus with boundary.

scholarly journals Legendrian contact homology and topological entropy

2019 ◽  
Vol 11 (01) ◽  
pp. 53-108 ◽  
Author(s):  
Marcelo R. R. Alves
Keyword(s):  
Growth Rate ◽  
Topological Entropy ◽  
Infinite Family ◽  
Contact Structures ◽  
Legendrian Knots ◽  
Contact Homology ◽  
Contact Manifolds ◽  
3 Dimensional ◽  
Reeb Flows ◽  
Legendrian Contact Homology

In this paper we study the growth rate of a version of Legendrian contact homology, which we call strip Legendrian contact homology, in 3-dimensional contact manifolds and its relation to the topological entropy of Reeb flows. We show that: if for a pair of Legendrian knots in a contact 3-manifold [Formula: see text] the strip Legendrian contact homology is defined and has exponential homotopical growth with respect to the action, then every Reeb flow on [Formula: see text] has positive topological entropy. This has the following dynamical consequence: for all Reeb flows (even degenerate ones) on [Formula: see text] the number of hyperbolic periodic orbits grows exponentially with respect to the period. We show that for an infinite family of 3-manifolds, infinitely many different contact structures exist that possess a pair of Legendrian knots for which the strip Legendrian contact homology has exponential growth rate.

scholarly journals Biquandle cohomology and state-sum invariants of links and surface-links

2018 ◽  
Vol 27 (11) ◽  
pp. 1843016
Author(s):  
Seiichi Kamada ◽  
Akio Kawauchi ◽  
Jieon Kim ◽  
Sang Youl Lee
Keyword(s):  
Homology Theory ◽  
Oriented Surface ◽  
Marked Graph

In this paper, we discuss the (co)homology theory of biquandles, derived biquandle cocycle invariants for oriented surface-links using broken surface diagrams and how to compute the biquandle cocycle invariants from marked graph diagrams. We also develop the shadow (co)homology theory of biquandles and construct the shadow biquandle cocycle invariants for oriented surface-links.

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