c Pleiades Publishing, Ltd., 2012. ISSN 1061-9208, Russian Journal of Mathematical Physics, Vol. 19, No. 3, 2012, pp. 394–400. 

Normal Forms of Twisted Wire Knots A. B. Sossinsky Independent University of Moscow Institute for Problems in Mechanics Received Jume 26, 2012

Abstract. This article is a continuation of the study of new types of knot energy undertaken in [1, 2] (but is formally independent of those articles); it describes some experiments with mechanical models of knots (that we call twisted wire knots), contains rigorous definitions of their mathematical counterparts, formulations of a series of problems and conjectures. Different energy functionals for various classes of knot types and the corresponding normal forms are discussed and compared. DOI 10.1134/S1061920812030119

In this paper, we describe certain kinds of mechanical devices, called twisted wire knots, and define their mathematical models. The devices are very thin knotted solid tori made of flexible and resilient wire of fixed length and width; they can be deformed by bending and twisting (but cannot be stretched or compressed). Experiments (described below in Section 1) show that, if deformed from their equilibrium position, they always return to their original shape. What is more, in all our experiments, whenever two such devices of the same kind (i.e., of the same length, thickness and made of the same material) were originally knotted so as to have the same knot type, the shape that they acquired was always the same up to mirror symmetry. We call this shape the (mechanical) normal form of the given twisted wire knot. This paper is, in a certain sense, the continuation of the research initiated in my paper [1] and our joint paper [2] with Oleg Karpenkov, but is formally independent of these publications. No special knowledge of knot theory is needed to read the present paper, except the following basic notions: knot diagram, number of crossings of a knot diagram, isotopy of knot diagrams, knot type, knot tables, Reidemeister moves. In Section 2 of the present paper, we define mathematical models of twisted wire knots as smooth closed curves in 3-space of fixed length (with certain restrictions on their curvature κ) supplied with certain types of energy functionals. We call these models twisted knots. The models that we define depend on one real parameter d > 0 that we call thickness and a real number τ ; physically, it corresponds to the thickness of the wire, and mathematically it is characterized by how far κ and 1/τ are bounded away from zero. To a minimum of the chosen functional (this minimum can in principle be found by gradient descent in the corresponding space of knots under consideration) there correspond a concrete curve in 3-space that may be regarded as the (geometrical) normal form of the given twisted knot. Conjecturally, this normal form is unique up to mirror symmetry for twisted knots of the same isotopy class with a small number of crossings (here isotopy means isotopy in the class of twisted knots, see the precise definition in Section 2). It would be naive to hope that this approach could yield unique geometric normal forms for classical knots (without any restrictions on curvature and with the usual definition of isotopy) obtained by gradient descent along some functional. Actually, most specialists in knot energy believe that no such functional exists, and there is no hope of solving the knot classification problem by using any specific knot energy. However, our approach might yield a solution of the knot classification problem for knot diagrams with not too many crossing: given two knot diagrams, we choose a value of the thickness d small enough for us to be able to represent both of these diagrams as twisted knots of thickness d and find their normal forms; then we conjecture that the two given diagrams will have the same knot type (in the classical sense) if and only if their normal forms coincide or are mirror images of each other. 394

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In Section 3, we recall various definitions of knot energy (in particular M¨ obius energy and the energy functional for flat knots studied in [2], which involves Euler elastics, see [3]) and discuss energy functionals that might be appropriate for twisted knots. The concluding Section 4 is devoted to possible further developments of our approach to knot energy, some conjectures are stated, the practical aspects of calculating minima of our functionals and graphically displaying the normal forms on the computer screen by discretizing the problem are discussed. I am grateful to Oleg Karpenkov for many fruitful discussions and to Mikhail Zelikin for an illuminating commentary to a talk that I gave at the Fomenko seminar at Moscow State University. This work was partially supported by the RFBR grant # 12-01-00748a. 1. TWISTED WIRE KNOTS We consider a solid cylinder C of length 1 and cross section diameter d made out of a flexible resilient material (d is very small but positive); we denote by L the axis of C. Flexibility means that one can bend the cylinder without breaking it (of course the the cylinder cannot be bent too strongly: the radius of curvature of L will remain bounded away from zero by around 2d) and twist it around the axis L by an angle α (of course this angle shouldn’t be too large, otherwise the cylinder will break, so we always assume that α is less than a certain constant). The material from which the cylinder is made is non-stretchable (its length and cross section diameter do not change). Resilience means that the material, if bent or twisted, will tend to straighten out (force the curvature of L to vanish) and untwist itself (force the twist angle α to vanish). Now let us place the cylinder on a horizontal table, tie it (bending it along a knot diagram drawn on the table) into an (almost) flat knot and bring together its extremities (disks of diameter d) so that the cylinder C lies flat on the table except near crossings, where the overpassing branch lifts off the table by d; keeping one extremity fixed, twist the other one by some admissible (i.e., less than the prescribed constant) angle α and glue the extremities together (the angle α can be positive or negative: an orientation of L is fixed, and it determines the “positive” direction of twisting). The device thus constructed (which is a thin solid torus lying on the table) will be called a d-thin α-twisted wire knot or wire knot for short. Once constructed and released, a twisted solid knot will move in a well-determined way, not necessarily remaining almost flat (sometimes it even jumps off the table) and quickly comes to a standstill. This is because the knot, when constructed, acquires a certain amount of potential energy (resistance against bending and torque due to twisting) and, when released, its potential energy is transformed into kinetic energy; the wire knot first moves in a well-determined way to its equilibrium position, and oscillates around this position with decreasing amplitude as friction transforms the kinetic energy into thermal energy until no energy remains, and the wire knot stops moving. Note that the experiments were conducted under ordinary conditions, i.e., in the presence of the Earth’s gravitational force, which also acts on the knot, and if the resilience (and the reaction to twisting) of the knot is weak, the gravitational attraction will tend to flatten out the knot on the table. The shape that the wire knot then acquires will be called its normal form. The normal form is, geometrically, an embedded solid torus in the half space above the table. The normal form obtained is not necessarily almost flat, it is usually three-dimensional (in the sense that parts of it may move to positions high above the table), and its surface often has self-tangencies (which may be points or lines on the surface of the solid torus). The normal form depends not only on d, α, and on the initial knot diagram, but also on the physical parameters of the material from which the wire is made: its resilience (resistance to bending) and its untwisting force. Figure 1 shows photographs of the normal forms of the same knot (namely the “eight knot”, standardly denoted by 41 in knot tables) obtained by using wires with different resilience and torque parameters. Note that both of the normal forms have four self-tangencies, but one is (almost) flat while the other is quite three-dimensional. The following observation is the result of thousands of experiments. Observation 1. (stable equilibrium). If a twisted solid knot in normal form is subjected to a small continuous deformation, then it always rapidly returns to the same normal form. Thus a wire RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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Fig. 1. Two normal forms of the 41 knot made with different kinds of wire.

knot in normal form is in a stable equilibrium position. Here “small” doesn’t mean very small. Thus, after a tiny deformation, the wire knots we worked with usually stayed put (did not return to normal form) because of frictional forces, and only a small but substantial deformation led to the return to normal form. Thus the above observation is true only if applied to a slightly idealized (almost frictionless) version of our wire knot model. Observation 2. (unstable equilibrium). Besides normal forms in stable equilibrium, some wire knots also possess unstable equilibrium positions. The observed unstable positions were almost flat positions in which the wire remained motionless (because of the gravitational force pushing it down to the table and the frictional forces preventing its sliding), but any small perturbation resulted in the immediate move of the knot to a stable equilibrium position (its normal form). An experiment illustrating this state of affairs is shown in the next figure. The 41 wire knot in normal form in Figure 2 (a) was manipulated by the experimenter, who lifted a part of the knot (look at (b)), then flipped that part over (see (c)), flattened it out (d), after which the wire knot, of its own accord, acquired a nearly circular shape (that of a closed three-strand braid with four crossings) and stayed put (e). But after a small perturbation (a little push by the experimenter) the wire knot jumped up from the table and reacquired its normal form (f). Note, however, that the part (shown in black) of the knot lifted off the table at the beginning of the experiment occupies a different position in the reacquired normal form. The next observation concerns the possible uniqueness of normal forms and was described in [1]. By uniqueness we mean that any two wire knots (with zero twisting number) corresponding to isotopic knot diagrams have identical (i.e., isometric) normal forms. Here, as above, the observation pertains to slightly idealized wire knots. Observation 3. (uniqueness of the normal form?) All the wire knots corresponding to knot diagrams with seven crossings or less and zero twist angle (performed with the same wire) displayed unique up to mirror symmetry normal forms. Thus for a small number of crossings no non-uniqueness was revealed. The experiments (thousands of them) were performed with only one type of wire (the one shown on the photograph on the left in Figure 1); a smaller number of experiments were made with the wire shown on the right in that figure, and they also revealed no non-uniqueness. There are, however, examples of wire knots made of other material, when the normal form is not unique. A striking one is a wire knot with over 50 self-tangencies (constructed by the architect Dmitry Kozlov) which has two normal forms. One of them lies very close to the surface of an ellipsoid, while the other is close to the surface of a hyperboloid! Pictures of these two positions of a wire knot appear on p.112 (Figure 8) in Kozlov’s article [4]. 2. TWISTED KNOTS In this section, we pass from experimental physics to mathematics and define the mathematical counterparts of wire knots. The knots considered are ordinary three-dimensional smooth knots RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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Fig. 2. Deformation and return to normal form of the 41 knot through its unstable equilibrium position

satisfying certain additional conditions and supplied with a real number (called the twist number) which plays a key role when we define “normalizing functionals” (see Sec. 3 below). A d-thin α-twisted knot (twisted knot for short) Kd,α is a pair (γ, α), where α ∈ R and γ is a C 2 -smooth embedding γ : S1 → R3 of fixed length 2π whose curvature κ(γ(s)), s ∈ S1 , is uniformly bounded above by 2d and which possesses the following material resistance property: dK (s, s ) > 2d =⇒ |γ(s) − γ(s )| > d ∀s, s ∈ S1 , where dK denotes the distance along the curve K and | · | is the Euclidean distance in R3 ; the parameters d and α are called thickness and twist number, respectively. At this stage, α is just a real number “written” on the curve. Two twisted knots Kd,α and Kd  ,α are called isotopic if d = d , α = α , and there exists a continuous family of smooth homeomorphisms ht : R3 → R3 such that h0 = id, h1 (K) = K  , and ht (K) is a twisted knot for all t ∈ [0, 1]. The material resistance property says that points far away from each other on the curve cannot come too close together in space, thus formalizing the fact that the physical model (wire knot) is a solid torus of thickness d which can have self-tangencies but cannot push one of its parts into another one or tear it apart; therefore crossing changes in the process if an isotopy are impossible. Note that in our definition of twisted knot, the knot itself is a curve, not a solid torus. But the restrictions on its curvature and the material resistance requirement ensure that the curve behaves like the core of a solid torus under isotopy, in particular the curve cannot acquire self-tangencies or perform crossing changes. Thus the definition of twisted knots can indeed be regarded as the mathematical model of twisted wire knots. The above definition of isotopy is not more restrictive than that of ordinary isotopy (i.e., it preserves knot types), because we have the following statement. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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Proposition. Two twisted knots with the same twist number are isotopic (in the sense of the above definition of the isotopy of twisted knots) if and only they are isotopic as ordinary threedimensional knots (in the sense of the usual definition of isotopy). Proof. The “if part” of the assertion of the proposition is obvious. The “only if part” can be obtained by a straightforward but very tedious argument based on the Reidemeister Lemma (see e.g. [5]). 2. ENERGY FUNCTIONALS As we mentioned above, the main idea underlying the introduction of knot energy functionals (first discerned by Moffat [6]) was, originally, the optimistic hope that the minimum of an appropriately chosen functional would be a well-defined real number (unique within the isotopy class of the given knot), that the minimal values for non-isotopic knots would differ, that knots corresponding to the same minimum would be isometric, thus providing a unique normal form for isotopy classes of knots and thereby solving the knot classification problem. This ambitious program was not realized (and probably never will be), nevertheless, crucial advances were made. In the work of Fukuharu [7], the chosen functional was that of Coulomb energy: a uniform electrical charge was assumed to be uniformly distributed along the knot and gradient descent along the corresponding energy of the knot (in a discretized version of the problem) was used to obtain the local minimum of this energy for given knots and find the corresponding normal forms. Computer experiments showed that a local minimum would always be reached by gradient descent, but the normal forms turned out to be non-unique within certain isotopy classes. Note that different parts of the knot, having electrical charges of the same sign, repulse each other and this is what makes self-tangencies and crossing changes impossible in the process of gradient descent, thereby ensuring that the knot undergoes an isotopy during that process. The next breakthrough in the study of knot energies is due to Jun O’Hara [8, 9], who defined the following energy functional in the space of knots:   2π  2π  1 1 ds dt − 4, (1) − 2 E0 (γ) := ||γ(s) − γ(t)||2 dK (γ(s), γ(t)) 0 0 where γ : S1 → R3 is a C 2 -smooth knot of length 2π, s and t are arc length parameters on S1 , || · || is the Euclidean norm (distance) in R3 , and dK is the distance mesured along the curve γ. For this functional, computer experiments showed that the minimal value of E0 obtained by gradient descent (discretized version) from any unknot was always zero and the corresponding curve (normal form) was the round circle of length 2π. Moreover, for knot diagrams with 9 crossings or less, similar experiments showed that the minimal value of E0 (obtained by gradient descent) classifies knots in the sense that isotopic knots (with 9 crossings or less) have the same minimal value of E0 and that these values for non-isotopic knots are different (see [9]). However, in the seminal paper [10] by Freedman et al, it was shown that there are infinitely many different (non isometric but isotopic) knots having the same value of that minimum, so that it makes no sense to speak of normal forms related to the O’Hara functional E0 . Note that the form of knots obtained by gradient descent are determined by using discretized models of knots, functionals, and the process of gradient descent. There are very beautiful animations presenting the evolution of knots to their normal forms, see [11], [12]. In our approach, as a first step in the study of wire knots, we considered (in [2], a joint paper with O.Karpenkov) what we called flat knots. Mathematically, they are simply knot diagrams under flat isotopy, i.e., under sequences of Ω2 and Ω3 Reidemeister moves (the Ω1 move is forbidden, and therefore flat isotopy classes are much smaller than ordinary isotopy classes, so the number of classes is much larger). Physically, they are wire knots with zero twist number lying on a table and constrained between the table and a glass pane parallel to it and slightly lifted (by 3d, where d is the thickness of the wire). The functional that we used was  2π κ2 (γ(s))ds, (2) E2 (γ) = R2 (γ) + U2 (γ) = R2 (γ) + 0

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where γ(s) is the equation (in the arc length parameter s) of the plane curve defining the knot diagram, R2 (γ) is the material resistance functional (defined via a fairly intricate construction based on the area of so-called alternating cycles of the given knot diagram, for the details see [2]), κ(P ) is the curvature of γ at the point P , so that the intergral U2 (γ) (which we call the uniformization functional) is a classical expression defining the so-called Euler elastics (see e.g. [3]). It is proved in [2] that the uniformization functional U2 has a unique stationary value in the regular homotopy class of any regular curve γ and to such value corresponds a curve which is either a circle (passed once or several times) or a Bernoulli lemniscate (passed once or several times). Further, it is shown that any codimension one regular homotopy (for the definition see [2]) does not allow crossing changes and thus does not change the flat isotopy class of the flat knot. The uniqueness and the shape of the normal forms of flat knots with respect to the functional E2 is the object of ongoing research. For (three-dimensional) twisted knots, in the particular case when the twist angle an integer multiple of 2π, the following family of normalizing functionals may be proposed: E3 (γ) = λ1 R3 (γ) + λ2 U3 (γ) + λ3 T (γ),

(3)

where the λi are real numbers, γ is a smooth curve in space satisfying the definition of twisted knot, R3 is the material resistance functional in space, U3 is a functional similar to U2 , but defined for curves in 3-space, and T is a twisting energy functional. In the three-dimensional case, the material resistance functional (or rather its discretization) is very easy to define: roughly speaking, R3 is equal to zero except when two non-adjacent vertices of the polygonal line γ come closer together than the length of the edges, in which case it immediately becomes huge (like a kind of δ-function). To define T , note that a twist angle α = 2kπ, k ∈ Z, defines a framing (ribbon) on the given knot (e.g., for k = 0 it is the so-called blackboard framing), and for T one can take he potential energy which produces a field of unit forces on the knot perpendicular to the ribbon. Functionals of this type and the corresponding normal forms will be studied elsewhere. 5. PERSPECTIVES In this section, we formulate some concrete problems and discuss the perspectives of their solution. Problem 1. Using the planar uniformization functional U2 and the results of [2], give a proof of the Whitney-Graustein theorem [13] on the classification of regular curves in the plane. Essentially, this involves proving that the lemniscates passed more than once are unstable equilibria positions of the curves with respect to the functional U2 . Problem 2. Using a discretized version of the planar uniformization functional U2 , write a computer program that shows, as an animation, how an arbitrary regular curve on the plane is taken by an appropriate regular homotopy to its normal from (a lemniscate passed once or a circle passed once or several times), thus giving a visual illustration of the proof of the Whitney-Graustein theorem on the classification of plane regular curves. The solution of these two problems is the object of a joint article (in preparation) by Serguey Avvakumov, Oleg Karpenkov, and the author. Problem 3. Using the functional E2 (see Eq.(2)) with the summand R2 as defined in [2], study its extrema and the normal forms of flat knots by gradient descent. Problem 4. Using the functional E3 (see Eq.(3)) with λ1 = 0, λ2 = λ3 = 1, find the normal form of the knot diagram given by the round circle with twist angle α = 2kπ for various values of k ∈ Z. Work on these two problems is only beginning. Problem 5. For the functional E3 (see Eq.(3)) in general form (for various values of the coefficients λi ), investigate its stationary values, its minima, and the corresponding normal forms. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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Problem 6. Study knots supplied with the functional E3 (see Equation(3)) with λ1 = λ2 = 1, λ3 = 0 (no twisting) and investigate their gradient descent to normal form. Such knots may be called knotted Euler elastics – it would be remarkable if a functional introduced by Daniel Bernoulli and studied by Leonard Euler might provide us with an approach to the solution of the knot classification problem! REFERENCES 1. A.B.Sossinsky, “Mechanical Normal Forms of Knots and Flat Knots,” Russ. J. Math. Phys. 18 (2), 216–226 (2011). 2. O.Karpenkov, and A.Sossinsky, “Energies of Knot Diagrams,” Russ. J. Math. Phys. 18 (4), 306–317 (2011). 3. Yu.S.Osipov, and M.I.Zelikin, “Higher-Order Euler Elastics and Elastic Hulls,” Russ. J. Math. Phys. 19 (2), 163–172 (2012). 4. D.Yu.Kozlov, “Knots and Links As Form-Generating Structures,” Mathematics and Modern Art: Proceedings of the First ESMA Conference, Springer Proceedings in Mathematics, 18, Editor: Claude Bruter, 105–115 (2012). 5. V.V.Prasolov, and A.B.Sossinsky, Knots, Links, Braids and 3-Manifolds (AMS Publ., Providence, R.I., 1997). 6. H.K.Moffat, “The Degree of Knottedness of Tangled Vortex Lines,” J.Fluid Mech. 35 (1), 117-129 (1969). 7. W.Fukuhara, “Energy of a Knot,” The Fˆete of Topology, Academic Press, 443–451 (1988). 8. Jun O’Hara, “Energy of a Knot,” Topology 30 (2), 241–247 (1991). 9. Jun O’Hara, Energy of Knots and Conformal Geometry, (World Scientific, 2003). 10. M.H.Freedman, and Z.-X.He, Z.Wang, “M¨ obius Energy of Knots and Unknots,” Ann. of Math. 139 (2), 1–50 (1994). 11. K.Ahara, Energy of Knots (video available on youtube.com). 12. Ying-Qing Wu, Polygonal Knot Energy (available at http://www.uiowa.edu/ wu/min/ming.html). 13. H.Whitney, “On Regular Closed Curves in the Plane,” Comp.Math. 4, 276–284 (1937).

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