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Isomorphism of the groups of Vassiliev invariants of Legendrian and pseudo-Legendrian knots


Tchernov, V (2003). Isomorphism of the groups of Vassiliev invariants of Legendrian and pseudo-Legendrian knots. Compositio Mathematica, 135(1):103-122.

Abstract

The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of C-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact R3 are canonically isomorphic. Recently we constructed the first examples of contact 3-manifolds where Vassiliev invariants of Legendrian and of framed knots are different. Moreover in these examples Vassiliev invariants of Legendrian knots distinguish Legendrian knots that are isotopic as framed knots and homotopic as Legendrian immersions. This raised the question what information about Legendrian knots can be captured using Vassiliev invariants. Here we answer this question by showing that for any contact 3-manifold with a cooriented contact structure the groups of Vassiliev invariants of Legendrian knots and of knots that are nowhere tangent to a vector field that coorients the contact structure are canonically isomorphic.

The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of C-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact R3 are canonically isomorphic. Recently we constructed the first examples of contact 3-manifolds where Vassiliev invariants of Legendrian and of framed knots are different. Moreover in these examples Vassiliev invariants of Legendrian knots distinguish Legendrian knots that are isotopic as framed knots and homotopic as Legendrian immersions. This raised the question what information about Legendrian knots can be captured using Vassiliev invariants. Here we answer this question by showing that for any contact 3-manifold with a cooriented contact structure the groups of Vassiliev invariants of Legendrian knots and of knots that are nowhere tangent to a vector field that coorients the contact structure are canonically isomorphic.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:contact manifolds; knots; Vassiliev invariants
Language:English
Date:2003
Deposited On:29 Nov 2010 16:26
Last Modified:05 Apr 2016 13:25
Publisher:London Mathematical Society
ISSN:0010-437X
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:10.1023/A:1021733206049
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1955165

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