Möbius Invariance of Knot Energy

A physically natural potential energy for simple closed curves in R3 is shown to be invariant under Möbius transformations. This leads to the rapid resolution of several open problems: round circles are precisely the absolute minima for energy; there is a minimum energy threshold below which knotting cannot occur; minimizers within prime knot types exist and are regular. Finally, the number of knot types with energy less than any constant M is estimated. Consider a rectifiable curve γ(u) in the Euclidean 3-space R, where u belongs to R or S. Define its energy by E(γ) = ∫∫ { 1 |γ(u)− γ(v)|2 − 1 D(γ(u), γ(v))2 } |γ̇(u)||γ̇(v)| du dv, where D(γ(u), γ(v)) is the shortest arc distance betweenγ(u) and γ(v) on the curve. The second term of the integrand is called a regularization (see [O1–O3, FH]). It is easy to see that E(γ) is independent of parametrization and is unchanged if γ is changed by a similarity of R. Recall that the Möbius transformations of the 3-sphere = R ∪∞ are the tendimensional group of angle-preserving diffeomorphisms generated by inversion in 2-spheres. The central fact of this announcement is: Möbius Invariant Property. Let γ be a closed curve in R. If T is a Möbius transformation of R ∪∞ and T (γ) is contained in R, then E(T (γ)) = E(γ). If T (γ) passes through ∞, the integral satisfies E(T (γ)) = E(γ)− 4. This simple fact (proved below), combined with earlier results proved in [FH], allows the rapid resolution of several open problems. Theorem A. Among all rectifiable loops γ: S → R, round circles have the least energy (E (round circle) = 4) and any γ of least energy parameterizes a round circle. 1991 Mathematics Subject Classification. Primary 57M25, 49Q10; Secondary 53A04, 57N45, 58E30. Received by the editors April 14, 1992 The first author thanks the San Diego Supercomputer Center for use of their facilities. The second and fourth authors were supported in part by NSF grant DMS-8901412. The third author was supported in part by NSF grant DMS-9006954 c ©1993 American Mathematical Society 0273-0979/93 $1.00 + $.25 per page 1 2 STEVE BRYSON, M. H. FREEDMAN, Z.-X. HE, AND ZHENGHAN WANG Theorem B. If K is a smooth prime (not a connected sum) knot, then there exists a simple closed rectifiable γK of knot type K with E(γK) ≤ E(γ) for all rectifiable loops γ which are topologically ambient isotopic to K. Theorem C. Any minimizer γK , as above, will enjoy some regularity. With an arc length parametrization, γK will be in C. Several results of [FH] can be improved quantitatively. Theorem D. If γ is topologically tame, let c([γ]) denote the (topological) crossing number of the knot type. We have 2πc([γ]) + 4 ≤ E(γ). (It was proved in [FH] that finite energy implies tame.) Since an essential knot must have three or more crossings, we obtain the following Corollary. Any rectifiable loop with energy less than 6π+4 ≈ 22.84954 is unknotted. Computer experiments of [A] as reported in [O3] and independently by the first author yield an essential knot (a trefoil) with energy ≈ 74. It may be estimated [S,T,W] that the number K(n) of distinct knots of at most n crossings satisfies 2 ≤ K(n) ≤ 2 · 24. Hence the number of knot types with representatives below a given energy threshold can also be bounded by an exponential. Theorem E. The number Ke(M) of isomorphism classes of knots which have representatives of energy less than or equal to M is bounded by 2(24−4/2π)(241/2π)M ≈ (0.264)(1.658) . In particular, only finitely many knot types occur below any finite energy threshold. Note that there are competing candidates for the exponent = −2 in the definition of E; for example, the Newtonian potential in R has exponent = −1. When the exponent is strictly larger than−3, finite values are obtained for smooth simple loops. Exponents smaller or equal to −2 yield energies which blow up as a simple loop γ begins to acquire a double point, thus creating an infinite energy barrier to a change of topology. Such a barrier would not exist for the Newtonian potential. We refer to [O1–O3] for detailed discussions. Similarity and Möbius invariance are, of course, special to the exponent −2. Proof of Theorem A. Let T be a Möbius transformation sending a point of γ to infinity. The energy E(T (γ)) ≥ 0 with equality holding iff T (γ) is a straight line. Apply the Möbius invariant property to complete the proof. Proof of Theorem B. In [FH] it is shown that for prime knot types K minimizers exist in the class of properly embedded rectifiable lines whose completion in R∪∞ represent K. According to the Möbius Invariance Property, such lines may be moved to a closed minimizer by any Möbius transformation T which moves the completed line off infinity. Sketch of Proof of Theorem C. Let γK be a closed minimizer in knot type K. An inversion argument shows that, for sufficiently small ε > 0, if γK meets a closed ball MÖBIUS INVARIANCE OF KNOT ENERGY 3 of radius ε, Bε, only in its boundary Sε, then γK∩Sε consists of (at most) one point. The idea is that if γK ∩ Sε is disconnected, inverting an arc of γK\Sε into Bε will lower energy while preserving the knot type. Thus there is a continuous projection from the ε-neighborhood of γK to γK given by “closest point” π: Nε(γK) → γK . We prove that the fibers π−1(pt) are all geometric planar disks of radius ε. The disjointness of these “normal” fibers to distance ε is equivalent to the existence of a continuously turning tangent to γk whose generalized derivative is in L∞. A detailed proof of Theorem C will appear elsewhere. Proof of Theorem D. Theorem 2.5 of [FH] gives the inequality c([γ]) ≤ c(γ) ≤ E(γ)/2π for proper rectifiable lines. According to the Möbius Invariance Property, the energy will increase by exactly 4 if a Möbius transformation is used to move the line off infinity and into closed position. Proof of Möbius Invariance Property. It is sufficient to consider how I, an inversion in a sphere, transforms energy. Let u be the arc length parameter of a rectifiable closed curve γ, u ∈ R/lZ. Let