Torus Knots Extremizing the Conformal Energy

April 1993

Denise Kim

Department of Physics University of Massachusetts Amherst, MA 01003 Rob Kusner School of Mathematics Institute for Advanced Study Princeton, NJ 08540

1. Introduction.

Recently Freedman, He and Wang [FHW], following work of O’Hara [O], introduced an energy E(Γ) for a simple closed curve Γ ⊂ ℝ^{3}. The functional E is continuous on each isotopy class of curves, and tends to infinity as Γ nears selfintersection. Moreover, E is “proper” on the set of all isotopy classes, in the sense that there are only finitely many knot types below a given energy level.

A useful geometric property of E is conformal invariance: if μ is a Möbius transformation of ℝ^{3} ∪ ∞, and μ(Γ) ⊂ ℝ^{3}, then E(μ(Γ)) = E(Γ). This can be used to prove [FHW] that each prime knot class has an energy minimizing representative (of differentiability class C^{1,1}), and that the round circle is the unique energy minimizer among all curves (E = 4). It is also a non-trivial result of [FHW] that, at least for C^{1,1} curves, the functional E is sufficiently smooth to have a “gradient” dE (see §3). Thus it becomes an interesting problem to find E-critical curves, that is, solutions to dE = 0.

In this note we construct the first explicit examples of knotted curves which are critical for E. Our basic observation (§2) is that the conformally invariant energy E extends naturally to simple closed curves Γ ⊂ ℝ^{m}. In particular, if Γ ⊂ S^{3} ⊂ ℝ^{4} and σ : S^{3} ⟶ ℝ^{3} ∪ ∞ is stereographic projection, then E(σ(Γ)) = E(Γ) provided σ(Γ) ⊂ ℝ^{3}. Then we use the principle of symmetric criticality (§3) to show that for each relatively prime pair of integers (p, q) there is a (p, q)-torus knot Γ_{p,q} ⊂ S^{3} which is critical for E. This curve Γ_{p,q} is a principal orbit for an isometric S^{1}- action on S^{3}. In the same way, we construct critical orbital links with more than one component.

The energies E(Γ_{p,q}) can be computed explicitly in terms of rational trigonometric integrals. This makes them ideal benchmarks to check the accuracy of computer experiments with various discretized energies. We mention one such discrete model in §4. The authors have implemented several programs using this model, run a number of experiments, and tabulated some of the results (§4), but clearly more work needs to be done. In this regard, we should mention that several Japanese mathematicians – and independently, S. Bryson – as noted in [BFHW], have also conducted computer experiments to seek nontrivial extrema.

It is well-known (see, for example, [S], p. 250) that torus knots are prime, and it is tempting to conjecture that the stereographic images σ(Γ_{p,q}) are the energy minimizers guaranteed to exist by [FHW]. However, we expect this conjecture is false when both p and q are large, for the following reason: we can view Γ_{p,q} as a

Typeset by AMS-TEX

p- or q-strand braid lying on the surface of a torus, and if both p and q are at least 4, energy can be saved when “extra strands” of Γ_{p,q} depart from the surface of the torus and push into the interiors of the complementary solid tori. Our most recent experiments (§5) suggest that Γ_{p,q} is a stable local minimum for E when p or q is less than 4, but not a minimizer otherwise, lending evidence for our conjecture (as qualified) above. The general E-stability question will be explored elsewhere [KS].

2. Conformal Invariance.

Suppose Γ ⊂ ℝ^{m} is a C^{1,1} simple closed curve, meaning Γ admits a parametrization Γ : S^{1} ↪ ℝ^{m} whose first derivative Γ′(z) is Lipschitz. (Weaker differentiability assumptions are possible – see §1 of [FHW] – but we find C^{1,1} suitable for our purposes.) Let

D(x, y) = min{∫_{x}^{y} |Γ′(z)| dz , ∫_{y}^{x} |Γ′(z)| dz}

denote the distance between Γ(x) and Γ(y) within Γ.

Definition/Proposition Given Γ as above, its (conformal) energy

E(Γ) = ∫∫_{S^{1}×S^{1}}[\frac{1}{|Γ(x) − Γ(y)|^{2}} − \frac{1}{D^{2}(x, y)}] |Γ′(x)| |Γ′(y)| dxdy

is finite and independent of the parametrization of Γ. Moreover, if μ is a Möbius transformation of ℝ^{m} ∪ ∞, then E(μ(Γ)) = E(Γ) provided μ(Γ) ⊂ ℝ^{m}.

Proof: The arguments, with only formal modifications to move from 3 to m dimensions, are essentially in §1 and §2 of [FHW]. In particular, concerning the conformal invariance, the first term in the integrand is pointwise invariant, whereas the second term in the integral is “intrinsic”, in the sense that, for any ε > 0, the asymptotic expansion

∫∫_{|x−y|≥ε} \frac{|Γ′(x)| |Γ′(y)|}{D^{2}(x, y)}dxdy = \frac{4π}{ε} − 4 + O(ε)

is invariant under all C^{1,1} diffeomorphisms of ℝ^{m}, not only Möbius transformations. (Note that the 4π/ε term here precisely cancels a corresponding singular term from the first integral. This is the “regularization” of E introduced in [O].) □

Corollary. If Γ ⊂ S^{m} ⊂ ℝ^{m+1} and σ : S^{m} ⟶ ℝ^{m} ∪ ∞ is stereographic projection, then E(σ(Γ)) = E(Γ) provided σ(Γ) ⊂ ℝ^{m}.

Proof: Stereographic projection always extends to a Möbius transformation of ℝ^{m+1} ∪ ∞. □

Examples. Consider the (p, q)-torus knot Γ_{p,q}^{r} ⊂ S^{3} (\sqrt{1 + r^{2}}) ⊂ ℝ^{4} = ℂ^{2} defined by

Γ_{p,q}^{r}(x) = ( \begin{matrix}r cos px \\ r sin px \\ cos qx \\ sin qx \end{matrix} ) = ( \begin{matrix}r e^{ipx} \\ e^{iqx} \end{matrix} ) .

Then the conformal energy of its stereographic image σ(Γ_{p,q}^{r}) ⊂ ℝ^{3} is

E (σ(Γ_{p,q}^{r})) = E(Γ_{p,q}^{r})

= (r^{2}p^{2} + q^{2}) ∫_{0}^{2π}dy ∫_{y−π}^{y+π}dx[\frac{1}{r^{2}|e^{ipx} − e^{ipy}|^{2} + |e^{iqx} − e^{iqy}|^{2}} − \frac{1}{D^{2}(x, y)}]

= 2π(r^{2}p^{2} + q^{2}) ∫_{−π}^{π} dθ[\frac{1}{r^{2}|e^{ipθ} − 1|^{2} + |e^{iqθ} − 1|^{2}} − \frac{1}{(r^{2}p^{2} + q^{2})θ^{2}} ]

= 2π(r^{2}p^{2} + q^{2}) ∫_{0}^{π/2} dφ[\frac{1}{r^{2} sin^{2} pφ + sin^{2} qφ} − \frac{1}{(r^{2}p^{2} + q^{2})φ^{2}} ].

3. Symmetric Criticality.

For each pair (p, q) of integers, there is an isometric action α_{p,q} of S^{1} on the unit sphere S^{3} ⊂ ℝ^{4} = ℂ^{2} defined by

α_{p,q} (e^{iθ}) ∗ ( \begin{matrix}z \\ w \end{matrix} ) = ( \begin{matrix}e^{ipθ} z \\ e^{iqθ} w \end{matrix} ) .

This action is effective when p and q are relatively prime. The case (p, q) = (1, 1) is the familiar Hopf action, whose orbits are great circles in S^{3}. In general, the α_{p,q}-orbits of the points \binom{1}{0} or \binom{0}{1} are p- or q-fold coverings of the z- or w-axis circles S_{z}^{1} or S_{w}^{1}, respectively, while the principal α_{p,q}-orbit which passes through the point

( \begin{matrix}e^{it} \frac{r}{\sqrt{1+r^{2}}} \\ e^{it} \frac{1}{\sqrt{1+r^{2}}} \end{matrix} )

for r > 0, is simply a rotation and rescaling of the (p, q)-torus knot Γ_{p,q}^{r} defined in §2. We can think of (r, t) as polar coordinates for the 2-dimensional orbit space S_{p,q} = S^{3}/α_{p,q}(S^{1}) of this action.

Lemma. The knot Γ_{p,q}^{r} is critical for E if and only if \frac{∂}{∂r} E(Γ_{p,q}^{r}) = 0 for some r > 0.

Proof: The only if direction is clear, and the if direction is a consequence of the “principle of symmetric criticality” (see, for example, [P] for a general discussion of this principle) as follows: Consider a C^{1,1} vector field V : S^{1} ⟶ ℝ^{4} and a variation Γ_{ε} = Γ + εV of Γ = Γ_{p,q}^{r}. According to [FHW] (Lemma 6.1), there is a unique linear functional dE such that

dE(V) = \frac{d}{dε} E(Γ_{ε})|_{ε=0},

for any V as above. (More formally, the Banach space of C^{1,1} vector fields along Γ is the tangent space to the Banach manifold of C^{1,1} curves, and dE is the gradient one-form of E, at Γ. Several explicit formulas for dE are given in §6 of [FHW], but its linearity is the only property we need here.) Suppose to the contrary that dE(V) ≠ 0. Since Γ is an orbit of the S^{1}-action α = α_{p,q}, we can average the push-forward (α(e^{iθ}))_{∗}V over S^{1} to obtain an α-invariant vector field Ṽ for which dE(Ṽ) ≠ 0 as well.

Now Ṽ induces a variation through α-orbits, so we can view Ṽ as tangent to the orbit space S = S_{p,q} at Γ. But

dE|_{S} = \frac{∂E}{∂r} dr + \frac{∂E}{∂t} dt = \frac{∂E}{∂r} dr,

because rotation of S by e^{it} leaves E unchanged, and thus dE|_{S}(Ṽ) = dE(Ṽ) ≠ 0 implies \frac{∂E}{∂r} ≠ 0 as required. □

We come next to our main result.

Theorem 1. For each relatively prime pair of integers p and q greater than 1, there is a (p, q)-torus knot Γ_{p,q} ⊂ S^{3} which is critical for the conformal energy E. The knot Γ_{p,q} is a principal orbit of the S^{1}-action α_{p,q} defined above.

Proof: The energy E(r) = E(Γ_{p,q}^{r}) = E (\frac{1}{\sqrt{1+r^{2}}} Γ_{p,q}^{r}) is a smooth function of r.

Because the orbit \frac{1}{\sqrt{1+r^{2}}} Γ_{p,q}^{r} converges to either a p- or q-fold covering of a round circle as r approaches ∞ or 0, and since p, q ≥ 2, we have lim_{r↓0} E(r) = lim_{r↑∞} E(r) =

+ ∞. Thus there is a finite value r_{0} > 0 minimizing E(r). By the previous lemma, the principal orbit Γ_{p,q} = _{\frac{1}{\sqrt{1+r_{0}^{2}}}} Γ_{p,q}^{r_{0}} is critical for E. □

In [FHW] the conformal energy of a k-component link (Γ_{1}, ... , Γ_{k}) is defined as a natural extension of the conformal energy of a knot:

E(Γ_{1}, ... , Γ_{k}) = ∑_{1≤i,j≤k} E(Γ_{i}, Γ_{j}) ,

where E(Γ_{i}, Γ_{i}) = E(Γ_{i}) as before, and the cross-terms (i ≠ j) are

E(Γ_{i}, Γ_{j}) = ∫∫_{S^{1}×S^{1}} \frac{|Γ′_{i}(x)| |Γ′_{j}(y)|}{|Γ_{i}(x) − Γ_{j}(y)|^{2}} dxdy .

If we consider links whose k components are each orbits of the S^{1}-action α_{p,q}, the principle of symmetric criticality applies: the orbital link (Γ_{1}, ... , Γ_{k}) is critical for E among all links if and only if it corresponds to an E-critical configuration of k distinct regular points in the orbit space S = S_{p,q}, that is, to a critical point of E restricted to S^{k∗} = S^{k} \ D (here D is the big diagonal Δ of S^{k} union the singular orbits). But E is a positive proper function on S^{k∗}, so it has a minimum, implying:

Theorem 2. There are infinitely many E-critical links with any number of components. □

We refer to a link L in S^{3} ⊂ ℂ^{2} whose component curves are orbits for the Hopf action α_{1,1} as a Hopf link. Note that L corresponds to a collection of points on S_{1,1} = S^{2}, which we take to be the 2-sphere of radius 1. By a direct integration, as in the example of §2, one finds:

Proposition. For a Hopf link L, the energy cross-term E(Γ_{i}, Γ_{j}) = 2π^{2} csc σ, where σ is the the angle between 2-planes in ℝ^{4} meeting S^{3} in the Hopf orbits Γ_{i} and Γ_{j}, or equivalently, where s = 2σ is the angular distance between the corresponding points in S^{2}. □

Remark. We observed with John Sullivan that this means the total energy of a k-component Hopf link is, up to a constant, simply the “Coulomb” energy for k point charges in ℝ^{3} constrained to lie on S^{2}.

Corollary. The Hopf link L_{2} = S_{z}^{1} ∪ S_{w}^{1} with E = 8 + 4π^{2} is critical for E.

Proof: Since L_{2} corresponds to a pair of antipodal points on S^{2}, it is the Eminimizer among 2-component Hopf links. (In fact, L_{2} minimizes among all nontrivial 2-component links with at least one component a round circle [FHW], §7.) Therefore dE restricted to S_{1,1}^{2∗} = S^{2} × S^{2} \ Δ vanishes at L_{2}, so the principle of symmetric criticality gives the result. □

Remark. One can also show that the minimum energy Hopf link L_{k} with k = 3 or 4 components corresponds to the obvious equilateral configuration of k points on S^{2}. (Their energies are E(L_{3}) = 12 + 8\sqrt{3}π^{2} ≈ 148.7572 and E(L_{4}) = 16 + 12\sqrt{6}π^{2} ≈ 306.1059, respectively.) On the other hand, there is a 4-component critical link L_{4,e} (with E = 16 + 8(1 + 2\sqrt{2})π^{2} ≈ 318.2805), corresponding to 4 points arranged in an “equatorial square” on S^{2}, which is not a local minimum for E. Observe that L_{4,e} lies on the surface of a torus (since, in the orbit space S^{2}, these 4 points lie on a circle), but that energy is lowered as a component leaves the surface (in S^{2}, as the 4 points move to the regular tetrahedral configuration). If we think of L_{4,e} as a “(4, 4)-torus link”, this phenomenon illustrates the conjectured behavior (mentioned in the Introduction) of an E-minimizing (p, q)-torus knot for p and q at least 4. (Further results on Hopf links are given in [KS].)

4. Discretization and Computer Experiments.

Given a C^{1,1} knot Γ : S^{1} = ℝ/2πℤ ⟶ ℝ^{m}, and a sufficiently large integer n, the polygon γ with vertices γ(h) = Γ(2πh/n) , 1 ≤ h ≤ n, is a knot isotopic to Γ, which we call a polygonal approximation to Γ. One possible discretized energy is simply E(γ), which unfortunately is infinite! It can be regularized, by deleting the contributions of adjacent edges, to an energy E^{∗}(γ) which has the advantage of an infinite barrier to changing isotopy type (but the disadvantage of being difficult to compute). Another discrete energy e(γ), defined below, is much easier to compute. It can be shown that both E^{∗}(γ) and e(γ) converge to E(Γ) as γ approaches Γ for n very large.

Definition. The discrete energy of a polygon with vertices γ : ℤ/nℤ ⟶ ℝ^{m} is

e(γ) = ∑_{1≤i≠} ∑_{j≤n}[\frac{1}{|γ(i) − γ(j)|^{2}} − \frac{1}{d^{2}(i, j)}] |γ′(i)| |γ′(j)|,

where

d(i, j) = min{ ∑_{i≤h<j} |γ(h) − γ(h + 1)| , ∑_{j≤h<i} |γ(h) − γ(h + 1)|}

and

|γ′(h)| = \frac{|γ(h − 1) − γ(h)| + |γ(h) − γ(h + 1)|}{2}.

Observe that if γ = γ_{p,q} is a polygonal approximation to the orbit Γ_{p,q} ⊂ S^{3}, then (by setting h = i − j) the double-sum reduces to

e(γ) = n |γ′(1)|^{2} ∑_{1<h≤n} [\frac{1}{|γ(1) − γ(h)|^{2}} − \frac{1}{d^{2}(1, h)}].

We used this in a simple search routine called Torus Knot Energy to produce the following table of approximate values for the conformal energies of our critical (p, q)-torus knots. (Our program, written in FORTRAN, is available from polyphemus.gang.umass.edu by anonymous ftp in the directory pub/knot/.) The third column gives the corresponding critical radius (r_{0}). Note, by comparing the first six lines of the table with the exact value E = 4 for the round circle, that the approximate turning angle (d^{∘}, in degrees) at each vertex gives a rough percentage bound for the underestimation by e = e(γ_{p,q}) of the true energy E(Γ_{p,q}). The number of vertices (n) in the polygonal approximation γ_{p,q} is given in the far right column. Although not recorded in our table, we checked our computed values by interchanging p and q: to the number of digits given, this led to the same e and reciprocal r_{0}, as expected. (Bryson reports that his experiments found a trefoil – that is, a (2, 3)-torus knot – with energy about 74; compare with lines 7 through 10 of the table. The sharpest value we have computed for e(γ_{2,3}) is 74.4111 with r_{0} around 1.8572078 at n = 92160. From the geometric progression of the values of e as we double n, it appears 74.4121 < E(Γ_{2,3}) < 74.4122.)

5. Further Experiments.

Very recently, the second author and John Sullivan computed the gradient de of the discretized energy e, and had Ken Brakke incorporate this into his Surface Evolver computer program (Version 1.89). Because e and de are each quadratic in n, convergence of the negative gradient flow to a critical knot or link is considerably slower than in our program Torus Knot Energy, which searches only among orbits; the practical size of n in the Surface Evolver is limited accordingly. Nevertheless, we were able to test the stability of Γ_{2,q} and Γ_{3,q} for q small, as well as find a nonorbital (4, 5)-torus knot with energy E ≈ 388, considerably less than E(Γ_{4,5}) ≈ 406 (see table). We hope to report further on this in future joint work with Brakke, Sullivan, and a group of students at the Five Colleges Geometry Institute.

Acknowledgement. We thank the School of Mathematics at the Institute for Advanced Study, as well as the Center for Geometry, Analysis, Numerics and Graphics (G.A.N.G.) of the University of Massachusetts at Amherst, and the Geometry Center of the University of Minnesota, for the use of their computer facilities; and Ken Brakke, Bruce Kleiner and John Sullivan, for their helpful comments. Thanks also to Charlie Gunn and Peter Norman for their interest in this project. We are grateful to Dick Palais and the referee for suggesting a number of improvements to the text, and to Mike Davis for pointing out the reference [S].

7

\begin{matrix}p & q & r_{0} & e & d^{∘} & & n \\ 1 & 1 & —— & 3.9608 & 1 & 360 \\ 1 & 1 & —— & 3.9804 & .5 & 720 \\ 1 & 1 & —— & 3.9902 & .25 & 1140 \\ 1 & 1 & —— & 3.9951 & .12 & 2880 \\ 1 & 1 & —— & 3.9975 & .06 & 5760 \\ 1 & 1 & —— & 3.9987 & .03 & 11520 \\ 2 & 3 & 1.85... & 74.353 & .68 & 1440 \\ 2 & 3 & 1.857.. & 74.383 & .34 & 2880 \\ 2 & 3 & 1.8572. & 74.397 & .17 & 5760 \\ 2 & 3 & 1.85720 & 74.405 & .09 & 11520 \\ 2 & 5 & 3.4384. & 130.69 & .34 & 5760 \\ 2 & 5 & 3.43849 & 130.70 & .17 & 11520 \\ 2 & 7 & 4.9479. & 184.89 & .42 & 5760 \\ 2 & 7 & 4.94799 & 184.92 & .21 & 11520 \\ 2 & 9 & 6.4322. & 238.43 & .55 & 5760 \\ 2 & 9 & 6.43225 & 238.47 & .27 & 11520 \\ 3 & 4 & 1.4868. & 204.19 & .21 & 5760 \\ 3 & 4 & 1.48687 & 204.20 & .11 & 11520 \\ 3 & 5 & 1.9677. & 264.20 & .27 & 5760 \\ 3 & 5 & 1.96779 & 265.26 & .14 & 11520 \\ 3 & 7 & 2.9015. & 383.48 & .48 & 5760 \\ 3 & 7 & 2.90157 & 383.52 & .24 & 11520 \\ 3 & 8 & 3.3588. & 441.48 & .47 & 5760 \\ 3 & 8 & 3.35885 & 441.62 & .23 & 11520 \\ 4 & 5 & 1.3409. & 406.11 & .29 & 5760 \\ 4 & 5 & 1.34094 & 406.14 & .14 & 11520 \\ 4 & 7 & 2.0210. & 599.52 & .40 & 5760 \\ 4 & 7 & 2.02106 & 599.56 & .20 & 11520 \\ 4 & 9 & 2.6867. & 787.80 & .52 & 5760 \\ 4 & 9 & 2.68677 & 787.85 & .26 & 11520 \\ 5 & 6 & 1.2623. & 686.65 & .35 & 5760 \\ 5 & 6 & 1.26234 & 686.69 & .18 & 11520 \\ 5 & 7 & 1.5265. & 823.83 & .40 & 5760 \\ 5 & 7 & 1.52659 & 823.88 & .20 & 11520 \\ 5 & 8 & 1.7899. & 959.15 & .46 & 5760 \\ 5 & 8 & 1.78991 & 959.20 & .23 & 11520 \\ 5 & 9 & 2.0513. & 1093.0 & .52 & 5760 \\ 5 & 9 & 2.05137 & 1093.1 & .26 & 11520 \\ 6 & 7 & 1.2131. & 1050.9 & .41 & 5760 \\ 6 & 7 & 1.21313 & 1050.9 & .21 & 11520 \\ 7 & 8 & 1.1793. & 1503.0 & .47 & 5760 \\ 7 & 8 & 1.17939 & 1503.1 & .24 & 11520 \\ 7 & 9 & 1.3604. & 1726.4 & .52 & 5760 \\ 7 & 9 & 1.36044 & 1726.5 & .26 & 11520 \\ 8 & 9 & 1.1548. & 2046.6 & .54 & 5760 \\ 8 & 9 & 1.15483 & 2046.7 & .27 & 11520 \end{matrix}

References

[BFHW] S. Bryson, M.H. Freedman, Z.X. He and Z. Wang, Mobius Invariance of Knot Energy. Bull. Amer. Math. Soc. 28, (1993) 99–103.

[FHW] M.H. Freedman, Z.X. He and Z. Wang, On the Energy of Knots and Unknots. Geometry Center Research Report GCG40.

[KS] R. Kusner and J. Sullivan, A Conformally Invariant, Nonlocal Energy for Embedded Submanifolds. In preparation.

[O] J. O’Hara, Energy of a Knot. Topology 30, (1991) 241–247.

[P] R. Palais, The Principle of Symmetric Criticality. Comm. Math. Physics 69, (1979) 19–30.

[S] H. Schubert, Knoten und Vollringe. Acta Math. 90, (1953) 131–286.