Geometry & Topology, Vol. 2 (1998) Paper no. 3, pages 31--64.

A natural framing of knots

Michael T Greene and Bert Wiest

Abstract. 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.

Keywords. Knot, link, knot invariant, framing, natural framing, torus knot, Cayley graph

AMS subject classification. Primary: 57M25. Secondary: 20F05.

DOI: 10.2140/gt.1998.2.31

E-print: arXiv:math.GT/9803168

Submitted to GT on 4 August 1997. Paper accepted 19 March 1998. Paper published 21 March 1998.

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Michael T Greene, Bert Wiest
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK.
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