New Invariants of Legendrian Knots

Chekanov, Yuri

doi:10.1007/978-3-0348-8266-8_45

Cited by 9 publications

(18 citation statements)

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“…The space of such knots, modulo reparametrization, is denoted by Leg(S 1 , R 3 ). We will refer to connected components of Leg(S 1 , R 3 ) (with respect to the quotient of the C ∞ topology) as Legendrian knot types 5 . When talking about classical or smooth knots, we simply mean that the Legendrian assumption is not made.…”

Section: Preliminariesmentioning

confidence: 99%

“…Such a diagram always has an admissible decomposition (or ruling) in the sense of [5]: the only two values of the Maslov potential (even though it's Z-valued) are 1 on the upper strands and 0 on the strands of the original braid β , thus all crossings are Maslov, and we may declare all of them switching (in the multi-component case, consider only proper crossings). This gives rise to a decomposition where the discs are nested in one another, so it's admissible.…”

Section: Remark 73mentioning

confidence: 99%

“…The existence of a ruling implies that L β is not a stabilized link type for any positive braid β [5]. Also, by a theorem of Fuchs [18], it implies that the diagram has an augmentation 13 (note this is by no means unique).…”

Section: Remark 73mentioning

confidence: 99%

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Contact homology and one parameter families of Legendrian knots

Kálmán

2005

Geom. Topol.

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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.

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Section: Preliminariesmentioning

confidence: 99%

Section: Remark 73mentioning

confidence: 99%

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Contact homology and one parameter families of Legendrian knots

Kálmán

2005

Geom. Topol.

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“…Chekanov [3] and Eliashberg [7] showed that there exist formally Legendrian isotopic knots in R 2 × R which are not Legendrian isotopic using Legendrian contact homology. Legendrian contact homology associates a differential graded algebra (DGA) to a Legendrian knot K. The DGA is generated by Reeb chords on K, that is, flow lines of R starting and ending on K and the differential is given by a holomorphic curve count in the symplectization of the contact manifold.…”

Section: Introductionmentioning

confidence: 99%

Legendrian contact homology in the product of a punctured Riemann surface and the real line

Björklund

2016

J. London Math. Soc.

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We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in P × R, where P is a punctured Riemann surface. As an application we show that, for any integer k and any homology class h ∈ H1(P × R), there are k Legendrian knots, all representing h, which are pairwise smoothly isotopic through a formal Legendrian isotopy, but which lie in mutually distinct Legendrian isotopy classes.

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“…Beginning in the late 1990's, two nonclassical invariants of such Legendrian knots emerged. The first is based on the contact homology theory of Eliashberg and Hofer [Eliashberg 1998], and was couched in combinatorial form by Chekanov [2002a]. This invariant is a differential graded algebra that counts certain holomorphic disks in the symplectization of ‫ޒ‬ 3 .…”

Section: Introductionmentioning

confidence: 99%