#### Volume 2, issue 1 (2002)

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Framed holonomic knots

### Tobias Ekholm and Maxime Wolff

Algebraic & Geometric Topology 2 (2002) 449–463
 arXiv: math.GT/0206190
##### 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 $n\le \frac{|m|}{2}$, 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
Primary: 57M27
Secondary: 58C25