Framed holonomic knots

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 $n \leq \frac{| m |}{2}$ , 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.

Keywords

framing, holonomic knot, Legendrian knot, self-linking number, Whitney index

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 58C25

References

Forward citations

Publication

Received: 11 December 2001

Revised: 17 May 2002

Accepted: 28 May 2002

Published: 30 May 2002

Authors

Tobias Ekholm
Department of Mathematics Uppsala University PO Box 480 751 06 Uppsala Sweden

Maxime Wolff
Département de Mathématiques et Informatique Ecole Normale Supérieure de Lyon 46 allée d’Italie 69364 Lyon Cédex 07 France

Volume 2, issue 1 (2002)

Tobias Ekholm and Maxime Wolff