Volume 2, issue 1 (1998)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
A natural framing of knots

Michael T Greene and Bert Wiest

Geometry & Topology 2 (1998) 31–64

arXiv: math.GT/9803168


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.

knot, link, knot invariant, framing, natural framing, torus knot, Cayley graph
Mathematical Subject Classification
Primary: 57M25
Secondary: 20F05
Forward citations
Received: 4 August 1997
Accepted: 19 March 1998
Published: 21 March 1998
Proposed: Cameron Gordon
Seconded: Joan Birman, Walter Neumann
Michael T Greene
Mathematics Institute
University of Warwick
United Kingdom
Bert Wiest
Mathematics Institute
University of Warwick
United Kingdom