Upper Bounds for the Writhing of Knots and the Helicity of Vector Fields Jason Cantarella, Dennis Deturck, and Herman Gluck to Joan Birman on Her 70th Birthday

The writhing number of a curve in Euclidean 3-space, introduced by Călugăreanu (1959-61) and named by Fuller (1971), is the standard measure of the extent to which the curve wraps and coils around itself; it has proved its importance for molecular biologists in the study of knotted duplex DNA and of the enzymes which affect it. The helicity of a vector field defined on a domain in Euclidean 3-space, introduced by Woltjer (1958b) and named by Moffatt (1969), is the standard measure of the extent to which the field lines wrap and coil around one another; it plays important roles in fluid mechanics, magnetohydrodynamics, and plasma physics. In this paper, we obtain rough upper bounds for the writhing number of a knot or link in terms of its length and thickness, and rough upper bounds for the helicity of a vector field in terms of its energy and the geometry of its domain. Then we describe the spectral methods which can be used to obtain sharp upper bounds for helicity and to find the vector fields which attain them. Theorem A. Let K be a smooth knot or link in 3-space, with length L and with an embedded tubular neighborhood of radius R. Then the writhing number Wr(K) of K is bounded by |Wr(K)| < 1 4 ( L R ) 4 3 . Theorem B. Let V be a smooth vector field in 3-space, defined on the compact domain Ω with smooth boundary. Then the helicity H(V ) of V is bounded by |H(V )| ≤ R(Ω) E(V ) , where R(Ω) is the radius of a round ball having the same volume as Ω and the energy of V is given by E(V ) = ∫ Ω V · V d(vol). Theorem C. The helicity of a unit vector field V defined on the compact domain Ω is bounded by |H(V )| < 1 2 vol(Ω) 4 3 .