Volume 2, issue 1 (2002)

Download this article
For printing
Recent Issues

Volume 17
Issue 6, 3213–3852
Issue 5, 2565–3212
Issue 4, 1917–2564
Issue 3, 1283–1916
Issue 2, 645–1281
Issue 1, 1–643

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Framed holonomic knots

Tobias Ekholm and Maxime Wolff

Algebraic & Geometric Topology 2 (2002) 449–463

arXiv: math.GT/0206190


A holonomic knot is a knot in 3–space which arises as the 2–jet extension of a smooth function on the circle. A holonomic knot associated to a generic function is naturally framed by the blackboard framing of the knot diagram associated to the 1–jet extension of the function. There are two classical invariants of framed knot diagrams: the Whitney index (rotation number) W and the self linking number S. For a framed holonomic knot we show that W is bounded above by the negative of the braid index of the knot, and that the sum of W and |S| is bounded by the negative of the Euler characteristic of any Seifert surface of the knot. The invariant S restricted to framed holonomic knots with W = m, is proved to split into n, where n is the largest natural number with n |m| 2 , integer invariants. Using this, the framed holonomic isotopy classification of framed holonomic knots is shown to be more refined than the regular isotopy classification of their diagrams.

framing, holonomic knot, Legendrian knot, self-linking number, Whitney index
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 58C25
Forward citations
Received: 11 December 2001
Revised: 17 May 2002
Accepted: 28 May 2002
Published: 30 May 2002
Tobias Ekholm
Department of Mathematics
Uppsala University
PO Box 480
751 06 Uppsala
Maxime Wolff
Département de Mathématiques et Informatique
Ecole Normale Supérieure de Lyon
46 allée d’Italie
69364 Lyon Cédex 07