Knots and Their Curvatures

I discuss an old result of John Milnor stating roughly that if a closed curve in space is not too curved then it cannot be knotted. CONTENTS 1. The total curvature of a polygonal curve 1 2. A probabilistic interpretation of the total curvature 2 3. The total curvature of a smooth closed curve 4 4. Total curvature and knotting 6 References 7 1. THE TOTAL CURVATURE OF A POLYGONAL CURVE An (oriented) polygonal knot (or curve) is a closed curve C in R3, without selfintersections, obtained by successively joining n distinct points p1, . . . ,pn, pn+1 = p1 ∈ R via straight line segments [p1p2], . . . , [pn−1pn], [pn,p1]. The points pi are called the vertices of the polygonal knot C. We denote by VC the set of vertices. To each oriented edge [pi,pi+1], 1 ≤ i ≤ n, we associate the unit vector γi := 1 |−−−−→ pipi+1| · −−−−→ pipi+1. Denote by S the unit sphere in R3 centered at the origin. We obtain in this fashion a map γ = γC : VC → S, γ(pi) = γi. This is known as the Gauss map of the polygonal knot C. Let αi ∈ [0, π) be the angle between γi and γi+1; see Figure 1. We obtain in this fashion a map α = αC : VC → [0, π), α(pi) = αi. We define the total curvature of C to be the positive real number K(C) = 1 2π ∑ p∈VC αC(p) = 1 2π n ∑