Relative contact homology
- Relative contact homology
In mathematics, in the area of symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace. Namely, it is associated to a contact manifold and one of its Legendrian submanifolds. It is a part of a more general invariant known as symplectic field theory, and is defined using pseudoholomorphic curves.
Legendrian knots
The simplest case yields invariants of Legendrian knots inside contact three-manifolds. The relative contact homology has been shown to be a strictly more powerful invariant than the "classical invariants", namely Thurston-Bennequin number and rotation number (within a class of smooth knots).
Yuri Chekanov developed a purely combinatorial version of relative contact homology for Legendrian knots, i.e. a combinatorially defined invariant that reproduces the results of relative contact homology.
Tamas Kalman developed a combinatorial invariant for loops of Legendrian knots, with which he detected differences between the fundamental groups of the space of smooth knots and of the space of Legendrian knots.
Higher-dimensional legendrian submanifolds
In the work of Lenhard Ng, relative SFT is used to obtain invariants of smooth knots: a knot or link inside a topological three-manifold gives rise to a Legendrian torus inside a contact five-manifold, consisisting of the unit conormal bundle to the knot inside the unit cotangent bundle of the ambient three-manifold. The relative SFT of this pair is a differential graded algebra; Ng derives a powerful knot invariant from a combinatorial version of the zero-th degree part of the homology. It has the form of a finitely presented tensor algebra over a certain ring of multivariable Laurent polynomials with integer coefficients. This invariant assigns distinct invariants to (at least) knots of at most ten crossings, and dominates the Alexander polynomial and the A-polynomial (and thus distinguishes the unknot).
References
* Lenhard Ng, [http://front.math.ucdavis.edu/math.SG/0412330"Conormal bundles, contact homology, and knot invariants] .
* Tobias Ekholm, John Etnyre, Michael G. Sullivan, [http://front.math.ucdavis.edu/math.SG/0210124"Legendrian Submanifolds in $R^{2n+1}$ and Contact Homology"] .
* Yuri Chekanov, "Differential Algebra of Legendrian Links". Inventiones Mathematicae 150 (2002), pp. 441-483.
* [http://front.math.ucdavis.edu/math.GT/0407347 Contact homology and one parameter families of Legendrian knots by Tamas Kalman]
ee also
* Relative homology
Wikimedia Foundation.
2010.
Look at other dictionaries:
Contact geometry — Contact form redirects here. For a web email form, see Form (web)#Form to email scripts. The standard contact structure on R3. Each point in R3 has a plane associated to it by the contact structure, in this case as the kernel of the one form dz − … Wikipedia
Relative homology — In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways.… … Wikipedia
Floer homology — is a mathematical tool used in the study of symplectic geometry and low dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an… … Wikipedia
Homology modeling — Homology modeling, also known as comparative modeling of protein refers to constructing an atomic resolution model of the target protein from its amino acid sequence and an experimental three dimensional structure of a related homologous protein… … Wikipedia
Morse homology — In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be… … Wikipedia
List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… … Wikipedia
Floer-Homologie — Floer Homologien (FH) bezeichnet in der Topologie und Differentialgeometrie eine Gruppe ähnlich konstruierter Homologie Invarianten. Sie haben ihren Ursprung im Werk von Andreas Floer und sind seitdem ständig weiterentwickelt worden. Floer… … Deutsch Wikipedia
Floerhomologie — Floer Homologien (FH) bezeichnet in der Topologie und Differentialgeometrie eine Gruppe ähnlich konstruierter Homologie Invarianten. Sie haben ihren Ursprung im Werk von Andreas Floer und sind seitdem ständig weiterentwickelt worden. Floer… … Deutsch Wikipedia
Symplektische Feldtheorie — Floer Homologien (FH) bezeichnet in der Topologie und Differentialgeometrie eine Gruppe ähnlich konstruierter Homologie Invarianten. Sie haben ihren Ursprung im Werk von Andreas Floer und sind seitdem ständig weiterentwickelt worden. Floer… … Deutsch Wikipedia
evolution — evolutional, adj. evolutionally, adv. /ev euh looh sheuhn/ or, esp. Brit., /ee veuh /, n. 1. any process of formation or growth; development: the evolution of a language; the evolution of the airplane. 2. a product of such development; something… … Universalium
