Framed holonomic knots

Details Framed holonomic knots Framed holonomic knots

2002-06-18

arxiv.org/abs/math/0206190v1

Algebr. Geom. Topol. 2 (2002) 449-463

Mathematics

Geometric Topology

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-22.abs.html

Scientific paper

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.

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