Contact homology and one parameter families of Legendrian knots
By Tam'as K'alm'an. 2004
61 citations recorded by the source corpus.
We consider S 1 -families of Legendrian knots in the standard contact R 3 . We define the monodromy of such a loop, which is an automorphism of the Chekanov-Eliashberg contact homology of the starting (and ending) point. We prove this monodromy is a homotopy invariant of the loop (Theorem 1.1). We also establish techniques to address the issue of Reidemeister moves of Lagrangian projections of Legendrian links. As an application, we exhibit a loop of right-handed Legendrian torus knots which is non-contractible in the space Leg(S 1 , R 3 ) of Legendrian knots, although it is contractible in the space Emb(S 1 ,R 3 ) of smooth knots. For this result, we also compute the contact homology of what we call the Legendrian closure of a positive braid (Definition 6.1) and construct an augmentation for each such link diagram.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/4f3581055dacbac93caa058b2f6cb3d993284987
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In Discrete Mathematics and Combinatorics
- T. Ekholm
- Roger Casals
- Baptiste Chantraine
- Honghao Gao
- Frederick J. Bourgeois III
- M. Abouzaid
- A. Oancea
- Alexander F. Ritter
In Geometry and Topology
- Baptiste Chantraine
- Roger Casals
- Frederick J. Bourgeois III
- T. Ekholm
- Honghao Gao
- J. Sabloff
- L. Traynor
- Lenhard L. Ng
In Mathematical Physics
- Baptiste Chantraine
- T. Ekholm
- Frederick J. Bourgeois III
- Roger Casals
- Honghao Gao
- J. Sabloff
- L. Traynor
- Lenhard L. Ng
Citations in the corpus
- Infinitely many Lagrangian fillings (2020, background): “ts the Legendrian torus link ( n;m) = ( ) (R3;˘ st). Then the composition of the cyclic shift 2 exactly (n 1)-times yields a Legendrian loop 2n 2 for ( n;m). This is the Legendrian loop studied in [31] where it is shown to be a non-trivial Legendrian loop. We shall provide our own alternative proof ”
- Lagrangian cobordisms via generating families: Construction and geography (2014, background): “4 In [29], Kálmán constructed a loop of Legendrian trefoil knots — ie a spinnable isotopy — that is not contractible in the space of Legendrian trefoils; this isotopy was viewed in the context of spinning in [18].”
- Rational symplectic field theory over Z2 for exact Lagrangian cobordisms (2006, background): “In Subsection 2.3, the vector space of an exact cobordism is defined and gluings of formal disks are interpreted as operations on such vector spaces.”
- Contact homology and homotopy groups of the space of contact structures (2004, unknown): “ar X iv :m at h/ 04 07 53 1v 2 [ m at h. SG ] 2 1 Ju l 2 00”
- Lagrangian concordance of Legendrian knots (2006, background): “So before enlarging our current article, we plan to do two things: give a nontrivial example of such a map, and hopefully be able to compare this map with the one of [13] when the cylinder is constructed as in 3.”