1. We are indebted to W. Fenchel [5] for a theorem which is a left closed curve (in ordinary space) has total curvature ≥ 2π. Recently, K. Borsuk [3] gave a new proof of this theorem that applies to curves in R. In a note at the end of this paper, Borsuk asked the question wheter the total curvature of a left knotted curve is always ≥ 4π. The primary purpose of this note is to give an affirmati...

The proofs and applications are based on a Riemannian version of Gromov’s non-squeezing theorem and classical integral geometry. Given a convex surface Σ ⊂ R and a point q in the unit sphere S we denote by UΣ(q) the perimeter of the orthogonal projection of Σ onto a plane perpendicular to q. We obtain a function UΣ on the sphere which is clearly continuous, even, and positive. Let us denote the...

We define the total curvature of a semialgebraic graph Γ ⊂ R as an integral K(Γ) = R Γ dμ, where μ is a certain Borel measure completely determined by the local extrinsic geometry of Γ. We prove that it satisfies the Chern-Lashof inequality K(Γ) ≥ b(Γ), where b(Γ) = b0(Γ) + b1(Γ), and we completely characterize those graphs for which we have equality. We also prove the following unknottedness r...

This paper gives a partial confirmation of a conjecture of Agarwal, Har-Peled, Sharir, and Varadarajan that the total curvature of a shortest path on the boundary of a convex polyhedron in R3 cannot be arbitrarily large. It is shown here that the conjecture holds for a class of polytopes for which the ratio of the radii of the circumscribed and inscribed ball is bounded. On the other hand, an e...

The Fáry-Milnor Theorem says that any embedding of the circle S1 into R3 of total curvature less than 4π is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topological type of graphs Γ, what limitations on the isotopy class of Γ are implied by a bound on total curvature? Especially: what does “total curvature” mean for a graph? I shall disc...

The properties of double-stranded DNA and other chiral molecules depend on the local geometry, i.e., on curvature and torsion, yet the paths of closed chain molecules are globally restricted by topology. When both of these characteristics are to be incorporated in the description of circular chain molecules, e.g., plasmids, it is shown to have implications for the total positive curvature integ...

For a smooth closed surface C in E3 the classical total mean curvature is defined by M(C) = ¿/(«i + k2) do(p), where kx, k2 are the principal curvatures at p on C. If C is a polyhedral surface, there is a well known discrete version given by M(C) = IE/,(w a,), where 1¡ represents edge length and a, the corresponding dihedral angle along the edge. In this article formulas involving differentials...

We establish a new fundamental relationship between total curvature of knots and crossing number. If K is a smooth knot in R3, R the cross-section radius of a uniform tube neighborhood K, L the arclength of K, and κ the total curvature of K, then (up to some coefficient), crossing number of K ≤ L R κ . The proof generalizes to show that for smooth knots in R3, the crossing number, writhe, Möbiu...

The plasma membrane is a highly compartmentalized, dynamic material and this organization is essential for a wide variety of cellular processes. Nanoscale domains allow proteins to organize for cell signaling, endo- and exocytosis, and other essential processes. Even in the absence of proteins, lipids have the ability to organize into domains as a result of a variety of chemical and physical in...