A natural framing of knots

Details A natural framing of knots A natural framing of knots

1998-03-21

arxiv.org/abs/math/9803168v1

Geom. Topol. 2 (1998), 31-64

Mathematics

Geometric Topology

34 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper3.abs.html

Scientific paper

Given a knot K in the 3-sphere, consider a singular disk bounded by K and the intersections of K with the interior of the disk. The absolute number of intersections, minimised over all choices of singular disk with a given algebraic number of intersections, defines the framing function of the knot. We show that the framing function is symmetric except at a finite number of points. The symmetry axis is a new knot invariant, called the natural framing of the knot. We calculate the natural framing of torus knots and some other knots, and discuss some of its properties and its relations to the signature and other well-known knot invariants.

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