Algebraic & Geometric Topology

Framed holonomic knots

Tobias Ekholm and Maxime Wolff

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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.

Article information

Algebr. Geom. Topol., Volume 2, Number 1 (2002), 449-463.

Received: 11 December 2001
Revised: 17 May 2002
Accepted: 28 May 2002
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 58C25: Differentiable maps

framing holonomic knot Legendrian knot self-linking number Whitney index


Ekholm, Tobias; Wolff, Maxime. Framed holonomic knots. Algebr. Geom. Topol. 2 (2002), no. 1, 449--463. doi:10.2140/agt.2002.2.449.

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