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Symmetric Criticality for Tight Knots

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Symmetric Criticality for Tight Knots SYMMETRIC CRITICALITY FOR TIGHT KNOTS JASON CANTARELLA, JENNIFER ELLIS, JOSEPH H.G. FU, AND MATT MASTIN ABSTRACT. We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength critical configurations for knots and links which are different from the ropelength minima for these knot and link types. 1. INTRODUCTION Our goal in this paper is to investigate the shapes of knotted tubes of fixed radius and circular cross section with minimal length. This is known as the ropelength problem: minimize the length of a knotted or linked curve L subject to the constraint that the curve has “thickness one”. There are many equivalent characterizations of the unit thickness constraint[GM99, LSDR99, CFKS11], but the simplest uses Federer’s concept of “reach” [Fed59]: a curve has unit thickness if every point in a unit-radius neighborhood of the curve has a unique nearest neighbor on the curve. In previous work, some of us (Cantarella and Fu) developed a criticality theory for curves with a thickness constraint [CFK+06, CFKS11]. We gave a kind of Euler-Lagrange equation for curves which were min-critical for length when constrained by thickness which allowed us to characterize some families of critical curves explicitly and to solve explicitly for critical representatives of other link types (such as the Borromean rings). An old idea in analyzing geometric functionals is the principle of symmetric criticality as in the classic paper of Palais [Pal79]. The principle states that when a group G acts on a smooth manifold M by diffeo- morphisms, fixing a submanifold Σ of M and a smooth function f : M ! R, we expect that p 2 Σ will be critical for f on M if and only if p is critical for f on Σ. That is, we need only consider the effect of symmetric variations of p on f to decide whether all variations of p have directional derivative of f equal to zero. Palais gives hypotheses on G and M under which the principle is guaranteed to hold. The idea of this paper is to prove a similar symmetric criticality principle for tight knots. Although we are dealing with constrained criticality with a one-sided nonsmooth constraint, so that Palais’ theorems do not directly apply, the criticality theory from [CFKS11] will prove adaptable to this situation. We will then use our theory to analyze a few interesting examples. We first consider the case of (p; q) torus knots, showing arXiv:1208.3879v3 [math.DG] 20 Dec 2012 that any such knot has at least two distinct ropelength-critical configurations, one with p-fold and one with q-fold rotational symmetry. Coward and Hass [CH12] have recently shown that a split link composed of a connect sum of two nontrivial knots and an unknotted component has a configuration which cannot be isotoped to the global ropelength minimum for the link without strictly increasing the length of the unknotted component. We analyze similar configurations to show that they have mirror-symmetric ropelength critical configurations which are different from their ropelength global minima. Last, we consider an example due to Sakuma [Sak89] which we observe must have at least 4 non-congruent Z=2Z (2-fold rotationally symmetric) ropelength-critical configurations. Date: October 10, 2018. 1 2 JASON CANTARELLA, JENNIFER ELLIS, JOSEPH H.G. FU, AND MATT MASTIN FIGURE 1. The thickness of a smooth curve γ is controlled by curvature (as in the left picture), and chordlengths (as in the right picture). Since Hatcher’s proof of the Smale Conjecture [Hat83], we have known that the space of unknotted curves is contractible. It has long been conjectured that there is a natural “energy function” on unknots whose gradient flow realizes this contraction by evolving every unknot to the round circle [FHW94]. In particular, such an energy flow would be a natural unknot recognition algorithm. For the links in our examples, we show that the gradient flow of ropelength may encounter critical points before reaching the global minimum energy configuration, and hence that ropelength gradient flow is not a recognition algorithm for these knots and links. If these theorems could be extended to unknots, it would rule out ropelength (and, presumably, many other similar energies) as candidates for an unknot recognition flow. There is every reason to believe that there are ropelength critical unknots different from the round circle– Pieranski et. al. [PPS01] have given convincing numerical evidence, for example. Such configurations are often called “Gordian” unknots. We do not know yet whether symmetric criticality can help find Gordian unknots. It follows from work of Freedman and Luo [FL95] that for any subgroup G of O(3), the space of G-invariant unknots is path-connected and contains the round circle. However, Freedman and Luo do not prove that the space of G-invariant unknots is contractible. If there was a G so that the space of G-invariant unknots had a nonvanishing homotopy group, we could prove the existence of a configuration L of the unknot which was critical among G-invariant unknots. Our theorems below would then imply that L was ropelength-critical. 2. THICKNESS AND ITS FIRST VARIATION We begin with some definitions. Definition 1 (Federer [Fed59]). The reach of a closed space curve is the largest so that any point in an -neighborhood of the curve has a unique nearest neighbor on the curve. For space curves, we also refer to reach as half of the thickness of the curve. It is a classical theorem of Federer that curves with positive thickness are C1;1. Given this definition, we can define Definition 2. The ropelength Rop(L) of a curve L in R3 is the ratio of the length of the curve to its thickness. The ropelength Rop([L]) of a link type [L] is the minimum ropelength of all curves in the link type [L]. It is well known that ropelength minimizing curves or “tight knots” exist in every link type (see [GdlL03, GM99, CKS02, CFKS11] for variants of this theorem). We see in Figure 1 that the thickness of a curve L is controlled by both curvature and chordlengths (cf. [CFKS11]). In order to use our theory below, we need to make this intuition more precise (and deal with some technical issues). Following Figure 1, it would be reasonable to define the “chordlength” contribution to thickness in terms of local minima of the self-distance function d(x; y) = kx − yk on L × L. However, SYMMETRIC CRITICALITY FOR TIGHT KNOTS 3 this function has its global minimum of zero along the diagonal (where it is not differentiable), which is not the information we would like to capture. It turns out that nonzero local minima of self-distance are characterized by having the chord xy perpendicular to the tangent vector to L at both endpoints. We can preserve these lengths (but eliminate the problem of the diagonal) with a technical modification. Definition 3. Let (x; y) be the angle between the chord xy and the normal plane NxL to L at x. Given a link L, the penalized distance between two distinct points x; y 2 L is pd(x; y) := kx − yk sec2 (x; y): For y = x, we set pd(x; x) = 1. We say that the chord xy is in Strut(L) if pd(x; y) = Thi(L) and xy is in both NxL and NyL. The “curvature” contribution to thickness is harder to pin down, since the curves we are dealing with are 1;1 3 3 2 only C . Consider the space C3 of all oriented pointed circles in R , which we identify with R × TS by 3 2 taking (p; C) to correspond to (p; T; κ) 2 R × TS , where T is the oriented unit tangent to C at p and κ is its curvature vector there. Let R(p; T; κ) := 1=jκj 2 (0; 1] be the radius function. In this formulation the circles C may degenerate to lines, with κ = 0 and R = 1. Let Π be the projection Π:(p; C) 7! p. Given a C1;1 link L, by Rademacher’s theorem the set E on which the second derivative exists has full measure. Thus both the oriented unit tangent vector T (x) and the curvature vector κ(x) are well defined for x 2 E. We let Osc L ⊂ C3 be the set of all osculating circles: Osc L := x; T (x); κ(x) : x 2 E ⊂ C3: Its closure Osc L is a compact subset of C3 since jκj is bounded on E. Note that T = T (x) for any (x; T; κ) 2 Osc L, while of course κ ? T is some normal vector; thus we can view Osc L as a subset of the normal bundle to L. Definition 4. An osculating circle c 2 Osc L is said to be a kink if its radius R(c) is equal to Thi(L). 3 t We can define a variation ξ of a closed curve L 2 R to be the initial velocity field of a smooth family φ 2 3 3 0 for t 2 (−, ) of C diffeomorphisms of R ! R with φ = Id. The first derivative of the length of a strut as L is carried along by φt is easy to calculate. In [CFKS11], Lemma 4.2 , we show that φt induces a flow on the space of pointed circles C3 and compute the first derivative δξR(c) of the radius of a given circle at time 0 as a function of the variation field ξ.
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