Algebraic and Geometric Topology 2 (2002), paper no. 22, pages 449-463.

Framed holonomic knots

Tobias Ekholm, Maxime Wolff

Abstract. 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 2n < |m|+1, 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

AMS subject classification. Primary: 57M27. Secondary: 58C25.

DOI: 10.2140/agt.2002.2.449

E-print: arXiv:math.GT/0206190

Submitted: 11 December 2001. (Revised: 17 May 2002.) Accepted: 28 May 2002. Published: 30 May 2002.

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Tobias Ekholm, Maxime Wolff
Department of Mathematics, Uppsala University
P.O. Box 480, 751 06 Uppsala, Sweden
Departement de Mathematiques et Informatique, Ecole Normale Superieure de Lyon
46 allee d'Italie, 69364 Lyon Cedex 07, France


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