APPEARED IN BULLETIN OF THE

arXiv:math/9301212v1 [math.GT] 1 Jan 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 1, January 1993, Pages 99-103M¨OBIUS INVARIANCE OF KNOT ENERGYSteve Bryson, Michael H. Freedman,Zheng-Xu He, and Zhenghan WangAbstract. A physically natural potential energy for simple closed curves in R3is shown to be invariant under M¨obius transformations.This leads to the rapidresolution of several open problems: round circles are precisely the absolute minimafor energy; there is a minimum energy threshold below which knotting cannot occur;minimizers within prime knot types exist and are regular.

Finally, the number ofknot types with energy less than any constant M is estimated.Consider a rectifiable curve γ(u) in the Euclidean 3-space R3, where u belongsto R1 or S1. Define its energy byE(γ) =ZZ 1|γ(u) −γ(v)|2 −1D(γ(u), γ(v))2|˙γ(u)||˙γ(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]).

Itis easy to see that E(γ) is independent of parametrization and is unchanged if γ ischanged by a similarity of R3.Recall that the M¨obius transformations of the 3-sphere = R3 ∪∞are the ten-dimensional group of angle-preserving diffeomorphisms generated by inversion in2-spheres.The central fact of this announcement is:M¨obius Invariant Property. Let γ be a closed curve in R3.

If T is a M¨obiustransformation of R3 ∪∞and T (γ) is contained in R3, then E(T (γ)) = E(γ). IfT (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: S1 →R3, round circles have the leastenergy (E (round circle) = 4) and any γ of least energy parameterizes a roundcircle.1991 Mathematics Subject Classification. Primary 57M25, 49Q10; Secondary 53A04, 57N45,58E30.Received by the editors April 14, 1992The first author thanks the San Diego Supercomputer Center for use of their facilities.

Thesecond and fourth authors were supported in part by NSF grant DMS-8901412. The third authorwas supported in part by NSF grant DMS-9006954c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2STEVE BRYSON, M. H. FREEDMAN, Z.-X. HE, AND ZHENGHAN WANGTheorem B.

If K is a smooth prime (not a connected sum) knot, then there existsa simple closed rectifiable γK of knot type K with E(γK) ≤E(γ) for all rectifiableloops γ which are topologically ambient isotopic to K.Theorem C. Any minimizer γK, as above, will enjoy some regularity. With anarc length parametrization, γK will be in C1,1.Several results of [FH] can be improved quantitatively.Theorem D. If γ is topologically tame, let c([γ]) denote the (topological) crossingnumber of the knot type.

We have2πc([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 followingCorollary. Any rectifiable loop with energy less than 6π +4 ≈22.84954 is unknot-ted.Computer experiments of [A] as reported in [O3] and independently by the firstauthor 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 mostn crossings satisfies2n ≤K(n) ≤2 · 24n.Hence the number of knot types with representatives below a given energy thresholdcan also be bounded by an exponential.Theorem E. The number Ke(M) of isomorphism classes of knots which have rep-resentatives of energy less than or equal to M is bounded by 2(24−4/2π)(241/2π)M ≈(0.264)(1.658)M. In particular, only finitely many knot types occur below any finiteenergy threshold.Note that there are competing candidates for the exponent = −2 in the definitionof E; for example, the Newtonian potential in R3 has exponent = −1.

When theexponent is strictly larger than−3, finite values are obtained for smooth simpleloops. Exponents smaller or equal to −2 yield energies which blow up as a simpleloop γ begins to acquire a double point, thus creating an infinite energy barrier toa change of topology.

Such a barrier would not exist for the Newtonian potential.We refer to [O1–O3] for detailed discussions. Similarity and M¨obius invariance are,of course, special to the exponent −2.Proof of Theorem A.

Let T be a M¨obius transformation sending a point of γ toinfinity. The energy E(T (γ)) ≥0 with equality holding iffT (γ) is a straight line.Apply the M¨obius invariant property to complete the proof.□Proof of Theorem B.

In [FH] it is shown that for prime knot types K minimizersexist in the class of properly embedded rectifiable lines whose completion in R3∪∞represent K.According to the M¨obius Invariance Property, such lines may bemoved to a closed minimizer by any M¨obius transformation T which moves thecompleted line offinfinity.□Sketch of Proof of Theorem C. Let γK be a closed minimizer in knot type K. Aninversion argument shows that, for sufficiently small ε > 0, if γK meets a closed ball

M ¨OBIUS INVARIANCE OF KNOT ENERGY3of 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 willlower energy while preserving the knot type. Thus there is a continuous projectionfrom 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. Thedisjointness of these “normal” fibers to distance ε is equivalent to the existence ofa 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 inequalityc([γ]) ≤c(γ) ≤E(γ)/2πfor proper rectifiable lines.

According to the M¨obius Invariance Property, the energywill increase by exactly 4 if a M¨obius transformation is used to move the line offinfinity and into closed position.□Proof of M¨obius Invariance Property. It is sufficient to consider how I, an inversionin a sphere, transforms energy.

Let u be the arc length parameter of a rectifiableclosed curve γ, u ∈R/lZ. Let(1)Eε(γ) =ZZ|u−v|≥ε1|γ(u) −γ(v)|2 −1(D(γ(u), γ(v)))2du dvand(2)Eε(I ◦γ) =ZZ|u−v|≥ε1|I ◦γ(u) −I ◦γ(v)|2 −1(D(I ◦γ(u), I ◦γ(v)))2× ∥I′(γ(u))∥· ∥I′(γ(v))∥du dv.Clearly E(γ) = limε→0 Eε(γ) and E(I ◦γ) = limε→0 Eε(I ◦γ).It is a short calculation (using the law of cosines) that the first terms transformcorrectly, i.e.,∥I′(γ(u))∥· ∥I′(γ(v))∥|I(γ(u)) −I(γ(v))|2=1|γ(u) −γ(v)|2 .Since u is arclength for γ, the regularization term of (1) is the elementary integral(3)Z lu=0"2Z l/2v=ε1v2 dv#du = 4 −2lε .Let s be an arclength parameter for I ◦γ.

Then ds(u)/du = ∥I′(γ(u))∥where∥I′(γ(u))∥= f(u) denotes the linear expansion factor of I′. Since γ(u) is a lipschitzfunction and I′ is smooth, f(u) is lipschitz, hence, it has a generalized derivativef ′(u) ∈L∞.

(4)regularization (2) =Zu∈R/lZ"Z|v−u|≥ε|(I ◦γ)′(v)| dvD(I ◦γ(u), I ◦γ(v))2#|(I ◦γ)′(u)| du=ZR/lZ 4L −1ε+−1ε−ds,

4STEVE BRYSON, M. H. FREEDMAN, Z.-X. HE, AND ZHENGHAN WANGwhere L = Length(I(γ)) andε+ = ε+(u) = D((I ◦γ)(u), (I ◦γ)(u + ε)) = s(u + ε) −s(u)=Z u+εuf(w) dw = f(u)ε + ε2Z 10(1 −t)f ′(u + εt) dtandε−= ε−(u) = D((I ◦γ)(u −ε), (I ◦γ)(u)) = f(u)ε −ε2Z 10(1 −t)f ′(u −εt) dt.Since |f ′(w)| is uniformly bounded, we have1ε+=1f(u)ε"11 + (ε/f(u))R 10 (1 −t)f ′(u + εt) dt#=1f(u)ε1 −εf(u)Z 10(1 −t)f ′(u + εt) dt + O(ε2)=1f(u)ε −1f(u)2Z 10(1 −t)f ′(u + εt) dt + O(ε).Similarly,1ε−=1f(u)ε +1f(u)2Z 10(1 −t)f ′(u −εt) dt + O(ε).Then by (4)(5)regularization (2) = 4 −ZR/lZ2ε du+ZZR/lZ×[0,1](1 −t)f(u) [f ′(u + εt) −f ′(u −εt)] du dt + O(ε)= 4 −2lε + O(ε) + O(ε).Comparing (3) and (5), we getEε(γ) −Eε(I ◦γ) = O(ε);hence, E(γ) = E(I ◦γ).For the second assertion, let I send a point of γ to infinity.

In this case L = ∞and, thus, the constant term 4 in (5) disappears.□AcknowledgmentThe authors wish to thank Adriano Garsia and Fred Hickling for useful discus-sions.

M ¨OBIUS INVARIANCE OF KNOT ENERGY5References[A]K. Ahara, Energy of a knot, screened at Topology Conf., Univ. of Hawaii, August 1990,K.

H. Dovermann, organizer.[FH]M. H. Freedman and Z.-X.

He, On the ‘energy’ of knots and unknots (to appear). [O1]Jun O’Hara, Energy of a knot, Topology 30 (1991), 241–247.

[O2], Family of energy functionals of knots, Topology Appl. (to appear).

[O3], Energy functionals of knots (K. H. Dovermann, ed. ), World Scientific, Singapore(to appear).

[S]De Witt Sumners, The growth of the number of prime knots, Math. Proc.

CambridgePhilos. Soc.

102 (1987), 303–315.[T]W. T. Tutte, A census of planar maps, Canad.

J. Math.

15 (1963), 249–271.[W]D. J.

A. Welsh, On the number of knots, preprint. (Steve Bryson) Computer Sciences Corporation, NASA Ames Research Center,Moffett Field, California 94035E-mail address: bryson@nas.nasa.gov(Michael H. Freedman, Zhenghan Wang) Department of Mathematics, Universityof California at San Diego, La Jolla, California 92093-0112E-mail address: M. H. Freedman mfreedman@ucsd.edu(Zheng-Xu He) Department of Mathematics, Princeton University, Princeton,New Jersey 08544-1000Current address: Department of Mathematics, Cornell University, Ithaca, New York 14853

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