1998/03/21 by Michael T Greene, Michael T. Greene, Bert Wiest
Biochemistry, Genetics and Molecular Biology · Mathematics · #Bounded function #Connective tissue disorders research #Framing (construction) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Knot (papermaking) #Knot invariant #Knot theory #Mathematical analysis #Mathematics #Pure mathematics #Quantum invariant #Tricolorability #math.GT #msc:20F05 #msc:57M25
paper · pdf · doi:10.2140/gt.1998.2.31
published as Geom. Topol. 2 (1998) 31-64 · 34 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper3.abs.html
arxiv created 1998/03/21 · openalex publication_date 1998/03/21 · arxiv updated 2014/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.