scholarly journals A complete knot invariant from contact homology

2017 ◽  
Vol 211 (3) ◽  
pp. 1149-1200 ◽  
Author(s):  
Tobias Ekholm ◽  
Lenhard Ng ◽  
Vivek Shende
Keyword(s):  
Contact Homology ◽  
Knot Invariant

scholarly journals Framed cord algebra invariant of knots in S1 × S2

2015 ◽  
Vol 24 (14) ◽  
pp. 1550067
Author(s):  
Shawn X. Cui ◽  
Zhenghan Wang
Keyword(s):  
Main Tool ◽  
Finitely Generated ◽  
Contact Homology ◽  
The Third ◽  
Knot Invariant ◽  
Markov Theorem ◽  
Non Local

We generalize Ng’s two-variable algebraic/combinatorial zeroth framed knot contact homology for framed oriented knots in [Formula: see text] to knots in [Formula: see text], and prove that the resulting knot invariant is the same as the framed cord algebra of knots. Actually, our cord algebra has an extra variable, which potentially corresponds to the third variable in Ng’s three-variable knot contact homology. Our main tool is Lin’s generalization of the Markov theorem for braids in [Formula: see text] to braids in [Formula: see text]. We conjecture that our framed cord algebras are always finitely generated for non-local knots.

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.

Pseudo-cycles of surface-knots

2016 ◽  
Vol 25 (13) ◽  
pp. 1650068 ◽  
Author(s):  
Tsukasa Yashiro
Keyword(s):  
Cell Complex ◽  
Two Dimensional ◽  
Rectangular Cell ◽  
Knot Invariant

In this paper, we describe a two-dimensional rectangular-cell-complex derived from a surface-knot diagram of a surface-knot. We define a pseudo-cycle for a quandle colored surface-knot diagram. We show that the maximal number of pseudo-cycles is a surface-knot invariant.

scholarly journals Kontsevich’s integral for the Kauffman polynomial

1996 ◽  
Vol 142 ◽  
pp. 39-65 ◽  
Author(s):  
Thang Tu Quoc Le ◽  
Jun Murakami
Keyword(s):  
Finite Type ◽  
Knot Invariants ◽  
Chord Diagrams ◽  
Knot Invariant ◽  
Kauffman Polynomial ◽  
Zamolodchikov Equation

Kontsevich’s integral is a knot invariant which contains in itself all knot invariants of finite type, or Vassiliev’s invariants. The value of this integral lies in an algebra A0, spanned by chord diagrams, subject to relations corresponding to the flatness of the Knizhnik-Zamolodchikov equation, or the so called infinitesimal pure braid relations [11].

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 Contact homology and one parameter families of Legendrian knots

2005 ◽  
Vol 9 (4) ◽  
pp. 2013-2078 ◽  
Author(s):  
Tamas Kalman
Keyword(s):  
Legendrian Knots ◽  
Contact Homology

scholarly journals Legendrian contact homology and nondestabilizability

2011 ◽  
Vol 9 (1) ◽  
pp. 33-44 ◽  
Author(s):  
Clayton Shonkwiler ◽  
David Shea Vela-Vick
Keyword(s):  
Contact Homology ◽  
Legendrian Contact Homology
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