A natural framing of knots

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

Mathematical Subject Classification

Primary: 57M25

Secondary: 20F05

References

Forward citations

Publication

Received: 4 August 1997

Accepted: 19 March 1998

Published: 21 March 1998

Proposed: Cameron Gordon

Seconded: Joan Birman, Walter Neumann

Authors

Michael T Greene
Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom

Bert Wiest
Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom

Volume 2, issue 1 (1998)

Michael T Greene and Bert Wiest