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A natural framing of knots
Michael T Greene and Bert Wiest
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Geometry & Topology 2 (1998)
31–64
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Abstract
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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.
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Keywords
knot, link, knot invariant, framing,
natural framing, torus knot, Cayley graph
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Mathematical Subject Classification
Primary: 57M25
Secondary: 20F05
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Publication
Received: 4 August 1997
Accepted: 19 March 1998
Published: 21 March 1998
Proposed: Cameron Gordon
Seconded: Joan Birman, Walter Neumann
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