Geometry & Topology

A natural framing of knots

Michael T Greene and Bert Wiest

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513882901

Digital Object Identifier
doi:10.2140/gt.1998.2.31

Mathematical Reviews number (MathSciNet)
MR1608684

Zentralblatt MATH identifier
0891.57010

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

Keywords
knot link knot invariant framing natural framing torus knot Cayley graph

Citation

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. https://projecteuclid.org/euclid.gt/1513882901


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