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

top
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

top

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

top
  1. [1] V. I. Arnol'd, Plane curves, their invariants, perestroikas and classifications, in: Singularities and Bifurcations, V. I. Arnold (ed.), Adv. Soviet Math. 21, Amer. Math. Soc., Providence, 1994, 33-91. Zbl0864.57027
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.