Static Knot Energy, Hopf Charge, and Universal Growth Law Fanghua Lin Courant Institute of Mathematical Sciences New York University New York, New York 10021 and Yisong Yang Department of Mathematics Polytechnic University Brooklyn, New York 11201 Abstract We present a family of static knotted soliton energy functionals governing the configuration maps from the Euclidean space R4n−1 into the unit sphere S 2n so that the knot charges are naturally represented by the Hopf invariants in the homotopy group π4n−1 (S 2n) and the special case n = 1 recovers the classical Faddeev knot energy. We establish the general result that the minimum energy or the knot mass EN of knotted solitons of the Hopf charge N satisfies the universal fractional-exponent growth law EN ∼ |N |(4n−1)/4n, in which the fractional exponent depends only on the dimensions of the domain and range spaces of the configuration maps but does not depend on the detailed structure of the knot energy. PACS numbers: 11.27.+d, 11.10.Lm Key words: Faddeev knots, Hopf fibration, Skyrme energy, sublinear growth, Sobolev inequalities, knot energy, universality.

1

Introduction

It was Lord Kelvin who initiated the study of knots and explored the idea that atoms could be thought of made of knotted vortex tubes of ether so that the stability of matter could be explained by the topological stability of knots, the variety of chemical 1

elements could be viewed as a consequence of the variety of knots, and the spectral lines of atoms could be considered as a reflection of the oscillatory patterns of knots [1]. Although such an idea is now known to be incorrect, it has inspired various fundamental areas in modern science. For example, knots may be used to explain the concept of spin [2]; elementary particles may be regarded as quantized flux loops represented by knots or links and antiparticles by their mirror images [3]; knotted cosmic strings may be produced in the early stages of the universe which would be responsible for the initial matter accretion for galaxy formation [4, 5]; knotted structures may appear in ferromagnetic spin-triplet superconductors [6]; the topological classification of knots may give clues to various aspects of DNA [7]; the entanglement structures of polymers may also be understood based on a knowledge on knots [8]. Over the last one hundred years, numerous contributions to the classification of knots have been made. Notably, Tait [9] enumerated knots in terms of the crossing number of a plane projection; Alexander [10] discovered a knot invariant, known as the Alexander polynomial, arising in 3-dimensional homology; Jones [11] found a new knot invariant, known as the Jones polynomial, which enabled several conjectures of Tait to be proved [12]; based on a heuristic quantum-field theory argument, Witten [13] derived from the Chern–Simons action a family of knot invariants including the Jones invariant; finally came the Vassiliev invariants [14] which cover the Alexander polynomial and the Jones polynomial and lay a general framework for the study of the combinatorial aspects of knots. Recently, Faddeev and Niemi [15] used computer simulation, relaxation and toroidal coordinates to show that a ring-shaped (unknotted) Hopf charge one soliton exists as the energy minimizer of a relativistic quantum field theory model proposed many years ago by Faddeev [16], which may be viewed [17] as a refined Skyrme model [18] modeling mesons and baryons. A more extensive computer investigation was later conducted by Battye and Sutcliffe [19] and a variety of knotted solitons were obtained. In normalized form, the energy governing the static knotted configurations in the Faddeev model [15, 16, 19, 20] over the Euclidean 3-space R3 is Z n X o 1 X E(n) = |∂j n|2 + |Fjk (n)|2 dx, (1.1) 2 R3 1≤j≤3 1≤j
where the field n = (n1 , n2, n3 ) assumes its values in the unit 2-sphere and Fjk (n) = n · (∂j n ∧ ∂k n) (j, k = 1, 2, 3). The finite-energy condition implies that n approaches a constant vector at spatial infinity of R3 . Hence we may compactify R3 into S 3 and view the fields as maps from S 3 to S 2 . As a consequence, we see that each finiteenergy field configuration n is associated with an integer, Q(n), in π3(S 2 ) = Z. In fact, such an integer Q(n) is known as the Hopf invariant which has the following integral characterization due to Whitehead [21]: Since the vector field F = ( 21 jk` Fk` (n)) is divergence free, there is a vector potential A so that F = ∇ ∧ A. In terms of A and F, the Hopf charge Q(n) of the map n may then be evaluated by the integral Z 1 Q(n) = A · F dx, (1.2) 16π 2 R3 which is also a Chern–Simons invariant [22]. Mathematically, the knotted solitons of 2

the Faddeev model are the solutions to the minimization problem [23] EN = inf{E(n) | E(n) < ∞, Q(n) = N},

N ∈ Z.

(1.3)

It has been recognized [24] that the crucial characteristic that guarantees the existence of energy minimizing Faddeev knots is the sublinear energy growth law C1 |N|3/4 ≤ EN ≤ C2|N|3/4,

(1.4)

where C1 , C2 > 0 are universal constants. The left-hand side of (1.4) is known as the Vakulenko–Kapitanski lower bound [25] which prevents the field configurations of a nontrivial topological charge to collapse, and the right-hand side of (1.4) is derived in [24] which explains why knotted solitons of high topological charges are preferred over multiply distributed unknotted solitons. It is important to note that such a sublinear dependence relation between the energy and topology of knotted configurations has also been observed in other problems including large molecular conformation in polymers and gel electrophoresis of DNA. In these problems, a crucial geometric quantity that measures the “energy” of a physical knot of knot (or link) type K (or simply knot) is the “rope length” L(K), of the knot K. To define it, we consider a uniform tube centered along a space curve Γ. The “rope length” L(Γ) of Γ is the ratio of the arclength of Γ over the radius of the largest uniform tube centered along Γ. Then L(K) = inf{L(Γ) | Γ ∈ K}. A curve Γ achieving the infimum carries the minimum energy in K and gives rise to an “ideal” or “physically preferred” knot [26, 27], also called a tight knot [28]. Clearly, this ideal configuration determines the shortest piece of tube that can be closed to form the knot. Similarly, another crucial quantity that measures the geometric complexity of Γ is the average number of crossings in planar projections of the space curve Γ denoted by N(Γ) (say). The crossing number N(K) of the knot K is defined to be N(K) = inf{N(Γ) | Γ ∈ K}, which is a knot invariant. Naturally one expects the energy and the geometric complexity of the knot K to be closely related. Indeed, the combined results in [28, 29] lead to the relation C1 N(K)p ≤ L(K) ≤ C2N(K)p ,

(1.5)

where C1 , C2 > 0 are two universal constants and the exponent p satisfies 3/4 ≤ p < 1 so that in truly three-dimensional situations the preferred value of p is sharply at p = 3/4. This relation strikingly resembles the fractional-exponent growth law (1.4) for the Faddeev knots just discussed and reminds us once more that a sublinear energy growth law with regard to the topological content involved is essential for knotted structures to occur. In this paper, we establish that the sublinear energy growth law of the type (1.4) is in fact valid for the general Hopf fibration S 4n−1 → S 2n (n ≥ 1) where the knot energy is of the Faddeev type (1.1). We find the important universal fact that the fractional exponent at the lower and upper topological bounds is of the form (4n − 1)/4n where the numerator 4n−1 is the dimension number of the domain space and the denominator 4n is twice the dimension number of the range space, of the knot configuration maps, and is independent of the detailed structure of the knot energy. 3

2

Knot Energy and Hopf Charge

Let u : S 4n−1 → S 2n (n ≥ 1) be a differentiable map. Then there is an integer representation of u in the homotopy group π4n−1 (S 2n ), say Q(u), called R the generalized 2n 2n Hopf index of u. Let Ω be a volume form of S so that |S | ≡ S 2n Ω is the total volume of S 2n and u∗ the pullback map Λ(S 2n ) → Λ(S 4n−1 ) (a homomorphism between the rings of differential forms). Since u∗ commutes with d, we see that du∗ (Ω) = 0; since the de-Rham cohomology H 2n (S 4n−1 , R) is trivial, there is a (2n − 1)-form v ˜ = |S 2n |−1 Ω on S 4n−1 so that dv = u∗(Ω). Of course, the normalized volume form Ω ˜ Then, according to gives the unit volume and v˜ = |S 2n |−1 vRsatisfies d˜ v = du∗ (Ω). ∗ ˜ Whitehead, Q(u) may be represented as S 4n−1 v˜ ∧ u (Ω) or Z 1 Q(u) = 2n 2 v ∧ u∗(Ω). (2.1) |S | S 4n−1 Let ∗ be the Hodge dual induced from a fixed metric on S 4n−1 . Recall that the pointwise inner product h·, ·i on the space of p-forms at x, say Λpx , may be defined by α ∧ ∗β = hα, βi ∗ 1, α, β ∈ Λpx , and the norm defined by |α|2 = hα, βi. With ∗1 = dV denoting the canonical volume form of S 4n−1 , we introduce the generalized Faddeev knot energy over S 4n−1 as Z o n 1 (2.2) E(u) = |du|2 + |u∗(Ω)|2 dV. 2 S 4n−1 When we delete a point from S 4n−1 to obtain R4n−1 , the energy (2.2) leads us to the following Euclidean space energy Z n o 1 E(u) = |du|2 + |u∗(Ω)|2 dx, (2.3) 2 R4n−1 which coincides with that of Faddeev [15, 16, 23] when n = 1. Conversely, when we consider (2.3), we always restrict our attention to such a map u which is well behaved at the infinity of R4n−1 so that we can compactify R4n−1 to make u a well-defined map from S 4n−1 → S 2n . For such a map u, its Hopf index Q(u) is well defined and is given by Z 1 Q(u) = 2n 2 v ∧ u∗ (Ω), dv = u∗(Ω). (2.4) |S | R4n−1 It will be useful to note that the structure of u∗ (Ω) gives us the bound |u∗ (Ω)| ≤ C(n)|du|2n ,

(2.5)

where C(n) > 0 is a constant depending only on n.

3

Topological Lower Bound

We now derive the sublinear topological lower bound for the knot energy (2.3). We need the Sobolev inequality [30, 31, 32] over R4n−1 of the form C0 kfkq ≤ k∇fk2, 4

(3.1)

where k · kq denotes the Lq norm over R4n−1 and q satisfies 1 1 1 4n − 3 = − = . q 2 4n − 1 2(4n − 1)

(3.2)

It is clear that q lies in the range 2 < q ≤ 6 so that its conjugate exponent q 0 takes the value 2(4n − 1) q = (3.3) q0 = q−1 4n + 1 and lies in the range 6/5 ≤ q 0 < 2. In the representation (2.4), we can choose v to lie in the “Coulomb” gauge, d∗ v = 0, where d∗ denotes the adjoint of d. Therefore we have the direct relation ku∗ (Ω)k22 = kdvk22 = k∇vk22.

(3.4)

Now, in view of (3.1)–(3.4) and the H¨older inequality, we obtain Z 2n 2 |S | |Q(u)| ≤ |v| |u∗(Ω)| dx R4n−1

≤ kvk2(4n−1)/(4n−3) ku∗(Ω)k2(4n−1)/(4n+1) ≤ C0−1 ku∗(Ω)k2 ku∗ (Ω)k2(4n−1)/(4n+1).

(3.5)

The factor ku∗(Ω)k2 on the right-hand side of (3.5) can be controlled by the energy (2.3). Namely, ku∗ (Ω)k22 ≤ 2E(u). (3.6) In order to find a similar control for the factor ku∗(Ω)kq0 in (3.5) where q 0 is given by (3.3), we rewrite q 0 as q 0 = q1 + q2 and use the H¨older inequality to decompose ku∗(Ω)kq0 as Z 1/sq0  Z 1/tq0 ∗ ∗ q1 s ∗ q2 t 0 ku (Ω)kq ≤ |u (Ω)| dx |u (Ω)| dx , (3.7) R4n−1

R4n−1

where s, t > 1 and 1/s + 1/t = 1. We can choose 2 − q0 2(nq 0 − 1) , q2 = , 2n − 1 2n − 1 2n − 1 2n − 1 s = , t= 0 . 0 (2 − q )n nq − 1

q1 =

Note that q1s = 1/n, q2 t = 2. Thus, using (2.5) and (3.7), we arrive at Z 1/sq0  Z 1/tq0 ∗ 2 ku (Ω)kq0 ≤ C |du| dx |u∗(Ω)|2 dx . R4n−1

(3.8) (3.9)

(3.10)

R4n−1

Inserting (3.6) in (3.10) and noting that q 0 = 2(4n − 1)/(4n + 1), we see that (3.5) becomes 0

|S 2n |2|Q(u)| ≤ C1 E(u)1/q +1/2 = C1 E(u)4n/(4n−1). 5

(3.11)

Or equivalently, E(u) ≥ C|Q(u)|(4n−1)/4n,

(3.12)

where C > 0 depends only on n. In particular, when n = 1, we recover the classical 3/4-exponent growth law. Note that, with uλ (x) = u(λx), it is standard to verify the rescaling law Z n o λ E(uλ ) = λ3−4n |du|2 + |u∗(Ω)|2 dx, (3.13) 2 R4n−1 which says that the two energy terms respond to the rescaling in a similar fashion as those in the energy of the classical Faddeev model over R3 . In particular, if u is a critical point of the knot energy (2.3), we have (dE(uλ )/dλ)λ=1 = 0 or Z Z 1 2 (4n − 3) |du| dx = |u∗(Ω)|2 dx, (3.14) 2 R4n−1 R4n−1 which implies that the second term in the energy functional (2.3) becomes more dominant in higher dimensions.

4

Generalized Knot Energy

It will be interesting to note that the energy lower bound (3.12) is independent of the exponent in the kinetic energy density in (2.3). To see this, we replace (2.3) by Z o n 1 ∗ p 2 Ep (u) = |du| + |u (Ω)| dx, (4.1) 2 R4n−1 where p > 1 is an undetermined exponent. For the same q 0 given in (3.3), we write q 0 = q1 + q2 and we estimate the factor ku∗(Ω)kq0 in (3.5) again. Choose s and t so that p 2 − 2n s= 2 − q0

and

p 2 − 2n t= 0 p . q − 2n

(4.2)

Of course, formally, 1/s + 1/t = 1. Besides, the condition s > 1 simply implies that q 0 > p/2n, or 4n(4n − 1) . (4.3) 1 0 and q2 > 0 in (3.7) with q1 + q2 = q 0 so that   p 0 0 2 q − 2n p p(2 − q ) 2 q1 = = and q2 = = . (4.4) p 2ns 4n − p t 2 − 2n Inserting (4.4) into (3.7) and using (2.5), we have Z 1/sq0  Z ∗ p ku (Ω)kq0 ≤ C1 |du| dx R4n−1 1/q 0

≤ C2 Ep (u)

R4n−1

.

|u∗ (Ω)|2 dx

1/tq0 (4.5)

6

Substituting (4.5) into (3.5) and using (3.6) with E(u) replaced by Ep (u), we again find 0 |Q(u)| ≤ C3 Ep (u)1/2+1/q , (4.6) which is in the same form as (3.11). Hence Ep (u) ≥ C|Q(u)|(4n−1)/4n.

(4.7)

In other words, we cannot alter the fractional exponent (4n − 1)/4n in the lower bound estimate (4.7) by varying p in the permissible range (4.3) in the static knot energy (4.1). This is a remarkable and surprising feature of the knot energy (4.1) and the Hopf invariant (2.4). Shortly, we shall present a more surprising fact that a reversed version of (4.7) holds for the energy infimum so that the detailed structure of the knot energy functional is not important at all. Two special values of p in (4.1), p = 2n and p = 4n − 2, also recover the classical Faddeev energy (1.1) when n = 1 and may deserve some attention. For example, when p = 4n − 2, the knot energy becomes Z o n 1 ∗ 4n−2 2 E(u) = |du| + |u (Ω)| dx. (4.8) 2 R4n−1 This energy responds to the rescaling x → λx, uλ(x) = u(λx), exactly as in the classical Faddeev model over R3 , Z n o λ E(uλ) = λ−1 |du|4n−2 + |u∗ (Ω)|2 dx, (4.9) 2 R4n−1 so that in place of (3.14), we have the following energy partition identity, Z Z 1 4n−2 |du| dx = |u∗ (Ω)|2 dx, 2 R4n−1 R4n−1

(4.10)

which implies that the two energy terms in (4.8) make equal contributions to the total energy.

5

Topological Upper Bound

We now establish that a reversed inequality of the type (3.12) or (4.7) holds as well. In fact, we can show in this case that the detailed structure of the knot energy is not relevant and we reveal that the fractional exponent in the energy growth law only depends on the dimensions of the domain and range spaces of the configuration maps. For this purpose, we consider the energy functional of the general form Z E(u) = H(∇u) dx, (5.1) R4n−1

where H is a nonnegative-valued continuous energy density function satisfying H(0) = 0. Define EN = inf{E(u) | E(u) < ∞, Q(u) = N}. (5.2) 7

We shall extend the method in [24] to prove that there is a universal constant C > 0 such that EN ≤ C|N|(4n−1)/4n, N ∈ Z. (5.3) We first consider the case N = m2 , for a positive integer m. We decompose the upper hemisphere S+2n as S+2n = ∪m (5.4) i=1 B(i) ∪ D. Here B(i)’s are mutually disjoint geodesic balls of radius r = c0 /m1/2n inside S+2n where c0 > 0 is a small number independent of m. Note that this is where the dimension of the range space S 2n of the knot configuration maps comes into the picture for the first time when we consider the upper estimate of the energy infimum of solitons. We define a differentiable map v : S 2n → S 2n as follows: v(x) = (0, · · · , 0, 1) for all x ∈ S 2n \ ∪m i=1 B(i), and on each B(i), v is such that v|∂B(i) = (0, · · · , 0, 1), v(B(i)) covers S 2n exactly once, and v : B(i) → S 2n is orientation-preserving. In other words, the degree of the map from B(i) onto S 2n is exactly 1. We can further require that |∇v| ≤ c1 m1/2n for a positive constant c1 independent of m. We use Br (x0 ) to denote the ball in R4n−1 centered at the point x0 ∈ R4n−1 and of radius r and use Br to denote the ball centered at the origin and of radius r. We construct a map h : R4n−1 → S 2n such that h is a constant outside the ball Bm1/2n , |∇h| ≤ c2/m1/2n for a constant c2 independent of m, and that Q(h) = 1. For u = v ◦ h : R4n−1 7→ S 2n , we have Q(u) = (deg v)2Q(h) = m2 = N (see [33]). On the other hand, |∇u| ≤ c1 c2 and u(x) is a constant for x outside the ball Bm1/2n . Hence, for the energy defined in (5.1), we have Z (5.5) E(u) ≤ C1 dx = C1 |Bm1/2n | = C|N|(4n−1)/4n. |x|≤m1/2n

Note that this is where the dimension of the domain space R4n−1 of the knot configuration maps comes into the picture when we consider the upper estimate of the energy infimum of solitons. For the general case N ≥ 1, we have m2 ≤ N < (m + 1)2 for some positive integer m. We observe that ` = N − m2 < (m + 1)2 − m2 = 2m + 1 ≤ 3m.

(5.6)

Let h0 : B1 → S 2n be a smooth map with h0|∂B1 = (0, · · · , 0, 1) and Q(h0) = 1. Take ` points x1 , · · · , x` ∈ R4n−1 such that |xi | > m1/2n and that |xi − xj | > 1 + m1/2n for all i, j = 1, 2, · · · , `, i 6= j. We then define u˜ : R4n−1 → S 2n as follows:  for x ∈ Bm1/2n ,  u(x) = (v ◦ h)(x), h0 (x − xi ), for x ∈ B1 (xi ), i = 1, · · · , `, u˜(x) = (5.7)  (0, · · · , 0, 1), otherwise.

Note that u above is constructed as in the case N = m2 before. We have Q(˜ u) = Q(u) + ` = m2 + ` = N. 8

(5.8)

Using (5.6) and (5.7), we get E(˜ u) = E(u) + `E(h0 ) ≤ C1(m2 )(4n−1)/4n + 3mE(h0 ) ≤ CN (4n−1)/4n.

(5.9)

If N < 0, we simply need to change orientation. Consequently, we see that (5.3) is established. Hence, for the static knot energy (2.3) or (4.2), we see that the energy infimum EN evaluated over the configuration maps in the Hopf class Q = N satisfies the following fractional exponent growth law, C1 |N|(4n−1)/4n ≤ EN ≤ C2|N|(4n−1)/4n,

(5.10)

where C1 , C2 > 0 are universal constants. In summary, we have presented a class of static knot energy functionals which naturally realize the general Hopf fibration S 4n−1 → S 2n for arbitrary n and originate from the classical Faddeev knot energy in three dimensions. We found that the knot energy EN of the Hopf charge N obeys the universal sublinear growth law EN ∼ |N|(4n−1)/4n where the fractional exponent depends only on the dimensions of the domain and range spaces of the configuration maps but the detailed structure of the static knot energy is not important. FL was supported in part by NSF grant DMS–0201443. YY was supported in part by NSF grant DMS–0406446.

References [1] M. Atiyah, The Geometry and Physics of Knots, Cambridge Univ. Press, Cambridge 1990. [2] D. Finkelstein and J. Rubinstein, J. Math. Phys. 9 (1968) 1762. [3] H. Jehle, Phys. Rev. D 6 (1972) 441. [4] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects, Cambridge Univ. Press, Cambridge, 1994. [5] S. Perlmutter et al, Nature 391 (1998) 51. [6] E. Babaev, Phys. Rev. Lett. 88 (2002) 177002. [7] D. W. Sumners, Notices A. M. S. 42 (1995) 528. [8] A. MacArthur, in Knots and Applications, L. H. Kauffman (ed.), World Scientific, Singapore, 1995, p. 395. [9] P. G. Tait, Scientific Papers, Cambridge Univ. Press, Cambridge, 1900. [10] J. W. Alexander, Proc. Nat. Acad. Sci. USA 9 (1973) 93.

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[11] V. F. R. Jones, Bull. A. M. S. 12 (1985) 103; Ann. of Math. 126 (1987) 335. [12] K. Murasugi, Topology 26 (1987) 187; Knot Theory and its Applications, Birkh¨ auser, Boston, 1996. [13] E. Witten, Comm. Math. Phys. 121 (1989) 351. [14] V. A. Vassiliev, in Developments in Mathematics: the Moscow School, Chapman & Hall, London, 1993. p. 194. [15] L. Faddeev and A. J. Niemi, Nature 387 (1997) 58. [16] L. Faddeev, in Relativity, Quanta, and Cosmology, M. Pantaleo and F. De Finis (eds.), Johnson Preprint, New York, 1979. p.247. [17] Y. M. Cho, Phys. Rev. Lett. 87 (2001) 252001. [18] T. H. R. Skyrme, Proc. Roy. Soc. A 260 (1961) 127; ibid 262 (1961) 237; Nucl. Phys. 31 (1962) 556. [19] R. A. Battye and P. M. Sutcliffe, Phys. Rev. Lett. 81 (1998) 4798; Proc. Roy. Soc. A 455 (1999) 4305. [20] J. Hietarinta and P. Salo, Phys. Lett. B 451 (1999) 60. [21] J. H. C. Whitehead, Proc. Nat. Acad. Sci. U. S. A. 33 (1947) 117. [22] S. S. Chern and J. Simons, Proc. Nat. Acad. Sci. U. S. A. 68 (1971), 791; Ann. of Math. 99 (1974) 48. [23] L. Faddeev, Proc. Internat. Congress Mathematicians, Vol. I, Higher Ed. Press, Beijing, 2002, p. 235. [24] F. Lin and Y. Yang, Comm. Math. Phys. 249 (2004) 273. [25] A. F. Vakulenko and L. V. Kapitanski, Dokl. Akad. Nauk USSR 248 (1979) 810. [26] V. Katritch, J. Bednar, D. Michoud, R. G. Scharein, J. Dubochet, and A. Stasiak, Nature 384 (1996) 142. [27] B. Laurie, V. Katritch, Jose Sogo, T. Koller, J. Dubochet, and A. Stasiak, Biophys. J. 74 (1998) 2815. [28] J. Cantarella, R. B. Kusner, and J. M. Sullivan, Nature 392 (1998) 237. [29] G. Buck, Nature 392 (1998) 238. [30] T. Aubin, J. Diff. Geom. 11 (1976) 573. [31] G. Talenti, Ann. Mat. Pura Appl. 110 (1976) 352. [32] E. Lutwak, D. Yang, and G. Zhang, J. Diff. Geom. 62 (2002) 17. [33] D. Husemoller, Fibre Bundles, McGraw-Hill, New York, 1966.

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