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