Finite-type invariants of Legendrian knots in the 3-space: Maslov index as an order 1 invariant

Victor Goryunov; Jonathan Hill

Banach Center Publications (1999)

  • Volume: 50, Issue: 1, page 107-122
  • ISSN: 0137-6934

Abstract

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We consider a contractible closure of the space of Legendrian knots in the standard contact 3-space. We show that in this context the space of finite-type complex-valued invariants of Legendrian knots is isomorphic to that of framed knots in R 3 with an extra order 1 generator (Maslov index) added.

How to cite

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Goryunov, Victor, and Hill, Jonathan. "Finite-type invariants of Legendrian knots in the 3-space: Maslov index as an order 1 invariant." Banach Center Publications 50.1 (1999): 107-122. <http://eudml.org/doc/208999>.

@article{Goryunov1999,
abstract = {We consider a contractible closure of the space of Legendrian knots in the standard contact 3-space. We show that in this context the space of finite-type complex-valued invariants of Legendrian knots is isomorphic to that of framed knots in $R^3$ with an extra order 1 generator (Maslov index) added.},
author = {Goryunov, Victor, Hill, Jonathan},
journal = {Banach Center Publications},
keywords = {Legendrian curve; Legendrian knot; finite type invariant; plane front; Vassiliev invariants},
language = {eng},
number = {1},
pages = {107-122},
title = {Finite-type invariants of Legendrian knots in the 3-space: Maslov index as an order 1 invariant},
url = {http://eudml.org/doc/208999},
volume = {50},
year = {1999},
}

TY - JOUR
AU - Goryunov, Victor
AU - Hill, Jonathan
TI - Finite-type invariants of Legendrian knots in the 3-space: Maslov index as an order 1 invariant
JO - Banach Center Publications
PY - 1999
VL - 50
IS - 1
SP - 107
EP - 122
AB - We consider a contractible closure of the space of Legendrian knots in the standard contact 3-space. We show that in this context the space of finite-type complex-valued invariants of Legendrian knots is isomorphic to that of framed knots in $R^3$ with an extra order 1 generator (Maslov index) added.
LA - eng
KW - Legendrian curve; Legendrian knot; finite type invariant; plane front; Vassiliev invariants
UR - http://eudml.org/doc/208999
ER -

References

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  2. [2] V. I. Arnol'd, Topological Invariants of Plane Curves and Caustics, University Lecture Series 5, Amer. Math. Soc., Providence, 1994. 
  3. [3] V. I. Arnol'd, Invariants and perestroikas of plane fronts (in Russian), Trudy Mat. Inst. Steklov. 209 (1995), 14-64; English transl.: Proc. Steklov Inst. Mat. 209 (1995), 11-56. 
  4. [4] D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472. Zbl0898.57001
  5. [5] V. V. Goryunov, Vassiliev type invariants in Arnold’s J + -theory of plane curves without direct self-tangencies, Topology 37 (1998), 603-620. Zbl0909.57002
  6. [6] V. V. Goryunov, Vassiliev invariants of knots in R 3 and in a solid torus, in: Differential and Symplectic Topology of Knots and Curves, S. Tabachnikov (ed.), Amer. Math. Soc. Transl. Ser. 2, 190, Amer. Math. Soc., Providence, 1999, 37-59. Zbl0946.57017
  7. [7] V. V. Goryunov, Finite order invariants of framed knots in a solid torus and in Arnold’s J + -theory of plane curves, in: Geometry and Physics, J. E. Andersen, J. Dupont, H. Pedersen and A. Swann (eds.), Lecture Notes in Pure and Appl. Math. 184, Marcel Dekker, New York, 1997, 549-556. Zbl0871.57007
  8. [8] J. W. Hill, Vassiliev-type invariants of planar fronts without dangerous self-tangencies, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 537-542. Zbl0881.57003
  9. [9] M. Kontsevich, Vassiliev's knot invariants, in: I. M. Gel'fand Seminar, S. Gel'fand, S. Gindikin (eds.), Adv. Soviet Math. 16, Part 2, Amer. Math. Soc., Providence, 1993, 137-150. Zbl0839.57006
  10. [10] V. A. Vassiliev, Cohomology of knot spaces, in: Theory of Singularities and its Applications, V. I. Arnol'd (ed.), Adv. Soviet Math. 1, Amer. Math. Soc., Providence, 1990, 23-69. 

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