Geometry & Topology

A natural framing of knots

Michael T Greene and Bert Wiest

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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.

Article information

Geom. Topol., Volume 2, Number 1 (1998), 31-64.

Received: 4 August 1997
Accepted: 19 March 1998
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 20F05: Generators, relations, and presentations

knot link knot invariant framing natural framing torus knot Cayley graph


Greene, Michael T; Wiest, Bert. A natural framing of knots. Geom. Topol. 2 (1998), no. 1, 31--64. doi:10.2140/gt.1998.2.31.

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