Tangent-point energies and ropelength as Gamma-limits of discrete tangent-point energies on biarc curves

Using interpolation with biarc curves we prove Γ-convergence of discretized tangent-point energies to the continuous tangent-point energies in the C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{1}$\end{document}-topology, as well as to the ropelength functional. As a consequence, discrete almost minimizing biarc curves converge to minimizers of the continuous tangent-point energies, and to ropelength minimizers, respectively. In addition, taking point-tangent data from a given C1,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{1,1}$\end{document}-curve γ, we establish convergence of the discrete energies evaluated on biarc curves interpolating these data, to the continuous tangent-point energy of γ, together with an explicit convergence rate.


Introduction
The ropelength 1 of a closed arclength parametrized curve γ : R/LZ → R 3 is defined as the quotient of its length and thickness, Here, for variational considerations the thickness [γ] is most conveniently expressed following Gonzalez and Maddocks [15] -without any regularity assumptions on the curve γ -as [γ] := inf s =t =τ =s R(γ(s), γ(t), γ(τ )), where R(x, y, z) denotes the circumcircle radius of the three points x, y, z ∈ R 3 .Motivated by numerous applications in the Natural Sciences ropelength is used in numerical computations (see [11,2,10,18] and the references therein) to mathematically model long and slender objects such as strings or macromolecules that do not self-intersect.In fact, it was proved rigorously in [16,9] that a curve of finite ropelength is embedded and of class C 1,1 (R/LZ, R 3 ), which means that its curvature exists and is bounded a.e. on R/LZ.Moreover, a curve γ with positive thickness [γ] > 0 is surrounded by an embedded tube with radius equal to [γ] as shown in [16, Lemma 3], which justifies the use of the non-smooth quantity [•] as steric excluded volume constraint.
The minimization over all triples of curve points to evaluate thickness in (1.2) is costly, which lead to the idea to replace minimization by integration; see [15, p. 4773].One such integral energy is the tangent-point energy TP q (γ) := (R/LZ) 2 1 r q tp (γ(s), γ(t)) ds dt, q ≥ 2, ( where the circumcircle radius is now replaced by the tangent-point radius r tp (γ(s), γ(t)) = |γ(s) − γ(t)| 2 2 dist(γ(s) + Rγ (s), γ(t)) , (1.4) i.e., the radius of the unique circle through the points γ(s) and γ(t) that is in addition tangent to the curve γ at γ(s).Also this energy implies self-avoidance and has regularizing properties.It was shown in [33] that if TP q (γ) is finite for some q > 2, then γ is embedded and of class C 1,1− 2 q (R/LZ, R 3 ).Later, Blatt [5] improved this regularity to the optimal fractional Sobolev 2 regularity W 2− 1 q ,q (R/LZ, R 3 ), which actually characterizes curves of finite TP q -energy.The knowledge of the exact energy space was then used to establish continuous differentiability of the tangent-point energy [7, Remark 3.1], [34], and to find TP q -critical knots by means of Palais's symmetric criticality principle [14].Very recently, long-time existence for a suitably regularized gradient flow for TP q was shown via a minimizing movement scheme [21].
But the tangent-point energy was also used in numerical simulations.Bartels et al added a desingularized variant of the TP q -energy in [4,3] as a self-avoidance term to the bending energy to find elastic knots.The impressive simulations of Crane et al. in [36] use the TP q -energy as well to avoid self-intersections, a higher dimensional tangent-point energy allows for computations on self-avoiding surfaces; see [35].
In the present paper we address the mathematical question of variational convergence of suitably discretized tangent-point energies towards the continuous TP q -energy, as well as towards ropelength.To account for the tangential information encoded in the tangent-point radius in (1.3) on the discrete level we use biarcs, i.e., pairs of circular arcs as in [30,17,11], which on the one hand, can interpolate point-tangent data γ(s i ), γ (s i ) ∈ R 3 × S 2  for i = 1, . . ., n of a given arclength parametrized curve γ ∈ C 1 (R/LZ, R 3 ).Every biarc curve β consisting of n consecutive biarcs is therefore a C 1,1 -interpolant of the curve γ.On the other hand, every biarc curve produces point-tangent data on its own, namely the points q i and unit-tangents t i at every junction of two consecutive biarcs.In order to avoid degeneracies we restrict to those biarc curves β whose biarcs have lengths λ i that are controlled in terms of the curve's length L (γ) by means of the inequality L (γ) 2n ≤ λ i ≤ 2L (γ) n for i = 0, . . ., n − 1. (1.7) Let B n be the class of biarc curves β satisfying (1.7).Accordingly, we define in a parameter-invariant fashion the discrete tangent-point energy E n q for n ∈ N and q ∈ [2, ∞) on closed C 1 -curves γ as with the straight lines l(q i ) := q i + Rt i for i = 0, ..., n − 1.Notice that both TP q and E n q are invariant under reparametrization of the curves, and they have the same scaling behaviour, TP q (dγ) = d 2−q TP q (γ) and E n q (dγ) = d 2−q E n q (γ) for all d > 0. (1.9) We restrict to injective C 1 -curves that are parametrized by arclength, denoted as the subset C 1 ia to state our main results.Theorem 1.1 (Γ-convergence to tangent-point energy).For q > 2 and L > 0 the discrete tangent-point energies E n q Γ-converge to the tangent-point energy TP q on the space C 1 ia (R/LZ, R 3 ) with respect to the • C 1 -norm as n → ∞, i.e., (1.10) As an immediate consequence we infer the convergence of almost minimizers in a given knot class K of the discrete energies E n q to a minimizer of the continuous tangent-point energy TP q in the same knot class K.
Corollary 1.2 (Convergence of discrete almost minimizers).Let q > 2, L > 0, and K be a tame knot class and b n ∈ C Then γ is a minimizer of TP q in C * and lim n→∞ E n q (b n ) = TP q (γ).Furthermore, it holds that γ ∈ W 2− 1 q ,q (R/LZ, R 3 ).
Moreover, the discrete tangent-point energies can also be used to approximate the non-smooth ropelength functional R in the sense of Γ-convergence.
Also here we can state the convergence of almost minimizers to ropelength minimizing curves in a prescribed knot class, which could be of computational relevance for the minimization of ropelength.
To the best of our knowledge, the only known contributions on variational convergence of discrete energies to continuous knot energies are the Γ-convergence results of Scholtes.In [25] he proves Γ-convergence of a discrete polygonal variant of the Möbius energy to the classic Möbius energy introduced by O'Hara [22].This result was strengthened later by Blatt [6].In [26,27] he proved the Γ-convergence of polygonal versions of ropelength and of integral Menger curvature to ropelength and to continuous integral Menger curvature, respectively.It remains open at this point if stronger types of variational convergence such as Hausdorff convergence of sets of almost minimizers can be shown for the non-local knot energies treated here, as was, e.g., established in [28] for the classic bending energy under clamped boundary conditions.It would be also interesting to set up a numerical scheme for the discretized tangent-point energies E n q to numerically approximate ropelength minimizers, in comparison to the simulated annealing computations in [30,11] The present paper is structured as follows.In Section 2 we provide the necessary background on biarcs -mainly following Smutny's work [30].Section 3 is devoted to the convergence of the discretized energies E n q including explicit convergence rates; see Theorem 3.1.In Section 4 we treat Γ-convergence towards the continuous tangentpoint energies, as well as convergence of discrete almost minimizers, to prove Theorem 1.1 and Corollary 1.2.Finally, in Section 5 we prove Γ-convergence to the ropelength functional, Theorem 1.3 and convergence of discrete almost minimizers to ropelength minimizers, Corollary 1.4.In Appendix A we establish convergence of rescaled and reparametrized convolutions in fractional Sobolev spaces.Appendix B contains some quantitative analysis of general C 1 -curves, and in Appendix C we prove some auxiliary general results in the context of Γ-convergence.

Biarcs and Biarc curves
The discrete tangent-point energy defined in (1.8) of the introduction is defined on biarc curves, which are space curves assembled from biarcs, i.e., from pairs of circular arcs.In this section we first present the basic definitions and a general existence result due to Smutny [30,Chapter 4], before specializing to balanced proper biarc interpolations needed in our convergence proofs later on.Definition 2.1 (Point-tangent pairs and biarcs).Let T := R 3 × S 2 be the set of pointtangent data [q, t], where S 2 is the unit sphere in R 3 .
(ii) A biarc (a, ā) is a pair of circular arcs in R 3 that are continuously joined with continuous tangents and that interpolate a point-tangent pair ([q 0 , t 0 ] , [q 1 , t 1 ]) ∈ T × T .The common end point m of the two circular arcs a and ā is called matching point.
The interpolation is meant with orientation, such that t 0 points to the interior of the arc a and −t 1 points to the interior of the arc ā; see Figure 1.For two points q 0 , q 1 ∈ R 3 we set d := q 1 − q 0 and e := q 1 −q 0 |q 1 −q 0 | = d |d| , and define R(e) := 2 e ⊗ e − Id = 2ee T − Id, which is a symmetric, proper rotation matrix representing the reflection of t at the unit vector e.Moreover, for a point-tangent pair (i) Let C 0 be the circle through q 0 and q 1 with tangent t 0 at q 0 and let C 1 be the circle through both points with tangent t 1 at q 1 .If t 0 + t * 1 = 0, we denote the circle through q 0 and q 1 with tangent t 0 + t * 1 at q 0 by C + , if t 0 − t * 1 = 0, we denote the circle through both points with tangent t 0 − t * 1 at q 0 by C − ; see Figure 1 on the right.(ii) A point-tangent pair ([q 0 , t 0 ] , [q 1 , t 1 ]) ∈ T × T is called cocircular, if C 0 = C 1 as point sets.A cocircular point-tangent pair is classified as compatible, if the orientations of the two circles induced by the tangents agree, and incompatible otherwise.
Remark 2.3.For a point-tangent pair ([q 0 , t 0 ] , [q 1 , t 1 ]) ∈ T × T , the compatible cocircular case is equivalent to t 0 − t * 1 = 0.In this case, the circle C − is not defined.The incompatible cocircular case is equivalent to t 0 + t * 1 = 0, thus the circle C + is not defined.
The following central existence result of Smutny not only states that interpolating biarcs always exist, but it also characterizes geometrically the possible locations of the corresponding matching points depending on the type of the point-tangent pair.For the precise statement we denote for an arbitrary circle C through q 0 and q 1 the punctured set C := C \ {q 0 , q 1 }.Proposition 2.4.[30,Proposition 4.7] For a given point-tangent pair ([q 0 , t 0 ] , [q 1 , t 1 ]) ∈ T × T , we denote by Σ + ⊂ R 3 the set of matching points of all possible biarcs interpolating the point-tangent pair.Then: ) is cocircular, we distinguish between two cases: (a) If the point-tangent pair is compatible, then (b) If the point-tangent pair is incompatible, then Σ + is the sphere passing through q 0 and q 1 perpendicular to the circle C − without the points q 0 and q 1 .(iii) Σ + is a straight line passing through q 0 and q 1 without the two points if and only if t 0 = t 1 and t 0 , e = 0. (iv) Σ + is a plane through q 0 and q 1 without the two points if and only if t 0 = t 1 and t 0 , e = 0.
A particularly powerful interpolation is possible if the location of the matching point m ∈ Σ + of the biarc is roughly "in between" the points q 0 and q 1 .The following definition states this precisely for the relevant cases (i), (ii)(a), and (iii) of Proposition 2.4.
(iii) A biarc is called proper if it interpolates a proper point-tangent pair with a matching point m ∈ Σ ++ .
We call a biarc γ-interpolating and balanced if it interpolates a point-tangent pair where we indicate the dependence of matching point and location by the index h.Item (iv) of Definition 2.5 requires that the matching point m h bisects the segment connecting γ(s) and γ(s + h).That this is indeed possible for sufficiently small h is the content of the following result.Note that here and throughout the paper we use the periodic norm |s − t| R/LZ := min to measure distances in the periodic domain R/LZ.Moreover, for a continuous function f on [0, L] we denote by ω f : [0, L] → [0, ∞) its modulus of continuity, which satisfies ω f (0) = 0 and which can be chosen to be concave and non-decreasing.
If the point-tangent pair is not incompatible cocircular and γ (s) = γ (s + h) holds, it follows from Proposition 2.4 (i) and (ii) (a) that Σ h + is the circle C + .Hence, Σ h ++ is a circular arc between γ(s) and γ(s + h).Thus, the matching point m h can be chosen in + is a straight line as a consequence of Proposition 2.4 (iii), since we obtain γ (s), e h > 0 by dividing (2.4) through |γ(s + h) − γ(s)|, thus excluding case (iv) of Proposition 2.4.Moreover, with γ (s) = γ (s + h) we infer for the unit vector e h := γ(s + h) − γ(s)/|γ(s + h) − γ(s)| by means of (2.1) and (2.2) Thus, the vector γ (s)+(γ (s + h)) * is a positive multiple of the vector e h .In particular, γ (s) + (γ (s + h)) * has the same orientation as e h .According to Definition 2.5, Σ h ++ is in this case the line segment between γ(s) and γ(s + h).Therefore, the matching point m h in Σ h ++ can be also chosen such that |m h − γ(s)| = |γ(s + h) − m h |.So, we are finished with the proof once we have shown that the smallness condition on h excludes case (ii) (a) of Proposition 2.4.Indeed, suppose that the point-tangent pair was incompatible cocircular.Then and using (2.2) we can write This representation inserted in (2.5) leads to γ (s + h) − γ (s) = 2 e h , γ (s + h) e h and hence, By virtue of inequality (B.6) in Lemma B.2 we conclude which is equivalent to ω γ (h) ≥ 2 3 contradicting our assumption on h.
Glueing together finitely many interpolating biarcs in a C 1 -fashion produces biarc curves precisely defined as follows.
Definition 2.7.[30, cf.Definition 6.1] (i) A closed biarc curve β : J → R 3 is a closed curve assembled from biarcs in a C 1 -fashion where the biarcs interpolate a sequence ([q i , t i ]) i∈I of point-tangent tuples.J is a compact interval, I ⊂ N bounded, and the first and last point-tangent tuple coincide.The set of such biarc curves is denoted by Bn where n is the number of indices contained in I.
(ii) We call a closed biarc-curve proper if every biarc of the curve is proper.
(iii) A biarc-curve is γ-interpolating for a given curve γ ∈ C 1 ia (R/LZ, R 3 ), and balanced if every biarc of the curve is γ-interpolating and balanced.
Notice that the set B n of closed biarc curves satisfying (1.7) introduced in the introduction is a strict subset of Bn .
Under suitable control of partitions of the periodic domain we can prove the existence of proper, γ-interpolating, and balanced biarc curves in Lemma 2.9 below.

and only if for
Then there is some N ∈ N such that for all n ≥ N there exists a proper γ-interpolating and balanced biarc-curve β n interpolating the point-tangent pairs Proof.By means of the defining inequality (2.8) for the (c 3)), and we can choose N ∈ N so large that the inequalities As a consequence of Lemma 2.6, there exists for all n ≥ N and i = 0, . . ., n − 1 a proper γ-interpolating and balanced biarc interpolating the point-tangent pair ).Now, we assemble for i = 0, . . ., n − 1 these n biarcs as in Definition 2.7 and obtain a biarc-curve with the required properties.
From now on, whenever we write β n for a given curve γ ∈ C 1 ia (R/LZ, R 3 ), we mean a proper γ-interpolating and balanced biarc-curve obtained in Lemma 2.9.By λ n,i we denote the length of the i-th biarc of the curve β n .In general, the elements β n do not have the same length as the interpolated curve γ.However, Smutny showed in [30] that under certain assumptions the sequence of the lengths (L (β n )) n∈N of β n converges towards the length L (γ) of γ.The following lemma is an essential ingredient for that proof.
Lemma 2.10.Let γ ∈ C 1,1 ia (R/LZ, R 3 ) and (β n ) n∈N be a sequence of proper γ-interpolating and balanced biarc curves as in Lemma 2.9.Then Proof.We identify the periodic domain R/LZ with [0, L] and check that (c 1 − c 2 )distributed partitions of R/LZ satisfy Smutny's requirements in [30,Notation 6.2,6.3]apart from nestedness of the mesh.The latter, however, is not necessary in her proof; whence we can apply [30,Lemma 6.8] to conclude.Now we show that the lengths L (β n ) of proper γ-interpolating and balanced biarc curves β n converge towards the length L (γ) of γ.
Proof.This follows straight from [30, Corollary 6.9]; under the same preconditions as we verified in the proof of Lemma 2.10.
In order to address convergence of biarc curves β n to the interpolated curve γ we need to reparametrize β n for all n ∈ N such that those reparametrizations are defined on R/LZ like γ is.An explicit reparametrization function which maps the arclength parameters of γ at the supporting points of the mesh to the arclength parameters of β n is constructed in [30,Appendix A].With that we can show the C 1 -convergence of a reparametrized sequence of biarc curves to the interpolated curve γ.
Theorem 2.12.Let γ ∈ C 1,1 ia R/LZ, R 3 , and let (β n ) n∈N be a sequence of proper γ-interpolating and balanced biarc curves parametrized by arclength.Then for Proof.We want to apply [30,Theorem 6.13], where Smutny showed C 1 -convergence under certain assumptions.Additionally to the hypotheses checked before in the proof of Lemma 2.10, we need to show that the so called biarc parameters Λ n,i of the i-th biarc of the biarc curve β n , representable as (cf.[30,Lemma 4.13]) where the m n,i are the matching points of the i-th biarc, are uniformly bounded from below and from above.In other words, we have to prove that there exist two constants Λ min , Λ max such that Using the fact, that the biarc curves are balanced, i.e. |m n,i − γ(s n,i )| = |m n,i − γ(s n,i+1 )|, and that γ is parametrized by arclength, we can then estimate by means of (B.5) in Lemma B.2 in the appendix On the other hand, by [30,Lemma 5.6] we have where the constant hidden in the O(h 2 n,i )-term only depends on the curve γ and where Λn,i is given by Λ max as the necessary uniform bound on the biarc parameters.Therefore, [30,Theorem 6.13] is applicable and we obtain that B n → γ in C 1 as n → ∞.

Discrete energies on interpolating biarc curves converge to the continuous TP q -energy
For the central convergence result of this section, Theorem 3.1, we work with discrete tangent-point energies Ẽn q with the larger effective domain Bn (see Definition 2.7 (i)), instead of with E n q introduced in (1.8) of the introduction, whose effective domain B n is defined by the constraint (1.7).In other words, These discrete energies evaluated on a sequence (β n ) n∈N of proper γ-interpolating and balanced biarc curves converge with a certain rate to the continuous TP q -energy of γ if γ is sufficiently smooth.Some of the ideas in the proof of the theorem are based on [25, Proposition 3.1] by Scholtes.
Bn for all n ∈ N, be a sequence of proper γ-interpolating and balanced biarc curves interpolating the point-tangent data Then there exists an N ∈ N such that for every ε > 0 there is a constant Proof.Set Υ = 4 c 2 c 1 and define for i, j ∈ {0, . . ., n} the periodic index distance |i − j| n := min{|i − j|, n − |i − j|}.
We then decompose with ds dt, Step 2: Now, we give an upper bound for 2 dist(l(t),γ(s)) |γ(s)−γ(t)| 2 for all s, t ∈ R with s = t.Without loss of generality we assume t < s.Then there exists a number k = k(s, t) ∈ Z satisfying |t − s| R/LZ = |kL + t − s|.We use the periodicity of γ and where we assumed without loss of generality that kL+t < s for the integrals.Therefore, by means of (3.3) Applying the calculations above we can estimate the first term on the right-hand side of (3.2) from above by Step 3: By Lemma 2.10 we have max k=0,...,n−1 which takes care of the second term on the right-hand side of (3.2).
Step 4: The sequence (M n ) n is assumed to be (c 1 − c 2 )-distributed, so that in view of (2.8) for all n ∈ N and k = 0, . . ., n − 1.
Then, similarly as before, ) and t ∈ [s n,j , s n,j+1 ), which we consider also in Steps 5 and 6.In addition, we assume from now on that |i − j| n > Υ.
Step 5: In order to estimate A i,j , we initially estimate for arbitrary a, b ≥ 0 ) and use estimate (3.11) to find for s ∈ [s n,i , s n,i+1 ) and t ∈ [s n,j , s n,j+1 ) Furthermore, combining (3.4) with (3.10) yields Hence, Moreover, we estimate again by virtue of (3.10) now for s For arbitrary τ ∈ R the mapping P γ (τ ) : R 3 → Rγ (τ ) defined as is the orthogonal projection onto the subspace Rγ (τ ) since |γ | = 1, and we have Moreover, we have for any τ, σ ∈ R Furthermore, we calculate for s ∈ [s n,i , s n,i+1 ) and t ∈ [s n,j , s n,j+1 ) using the linearity of the projection which in turn by means of (3.19) and (3.17) can be bounded from above by The last summand is bounded by Inserting (3.14) and (3.20) into (3.12)yields (3.21)In order to obtain an estimate for the denominator of A i,j we consider Step 6: To estimate B i,j , we use (3.11) and twice (3.10) leading to Thus, by (3.13), (3.24), and (3.22), ds dt Step 7: The expression n Step 8: Since γ ∈ C 1,1 , by [30,Lemma 5.8] Accordingly, there exists an N ∈ N and a constant c N ∈ (0, ∞) such that holds for all n ≥ N .By (3.5) and (3.7) we obtain from (3.27) (3.29) Step 9: Inserting (3.6), (3.8), (3.26) and (3.29) into (3.2) yields }, which gives the desired result.

Γ-convergence to the continuous tangent-point energy
In the present section, we show that the continuous tangent-point energy TP q is the Γ-limit of the discrete tangent-point energies E n q as n → ∞ (see Theorem 1.1).As a consequence, we deduce that limits of discrete almost minimizers are minimizers of the continuous tangent-point energy; see Corollary 1.2.Γ-convergence.In order to prove Theorem 1.1 we need to verify the liminf and limsup inequality, see [8,Definition 1.5].Here, the liminf inequality is verified in a rather straightforward manner 4.1), whereas the proof of the limsup inequality requires more work; see Theorem 4.4 below.

Theorem 4.1 (Liminf inequality). Let γ, γ
Proof.We may assume that lim inf n→∞ E n q (γ n ) < ∞.Then there exists a subsequence we deduce γ n k ∈ B n k for all k ∈ N; see (1.8) in the introduction.Denote the pointtangent pairs that are interpolated by γ n k as (([q n k ,i , t n k ,i ] , [q n k ,i+1 , t n k ,i+1 ])) i=0,...,n k −1 , with q n k ,0 = q n k ,n k and t n k ,0 = t n k ,n k for each k ∈ N. Furthermore, we denote by a n k ,0 , . . ., a n k ,n k the arclength parameters satisfying for all i = 0, . . ., n k − 1. Define for all s, t ∈ R/LZ with s = t the function where χ A denotes the characteristic function of a set A ⊂ R/LZ × R/LZ.Easy calculations show that for all s = t.
The functions f n k are non-negative, and measurable since they are piecewise constant.
Rewriting the discrete tangent-point energies as ds dt, allows us to apply Fatou's lemma to obtain the desired liminf inequality.
An important first ingredient in the proof of the limsup inequality is the use of convolutions that approximate γ in the C 1 -norm.Here, η ∈ C ∞ (R) is a non-negative mollifier with supp η ⊂ [−1, 1] and R η(x) dx = 1, and for any ε > 0 we set η ε (x) := 1 ε η x ε .In general, the convolutions are not parametrized by arclength even if γ is, and they do not need to have the same length as γ.Thus, we rescale the convolutions to have the same length as γ and reparametrize then according to arclength.The following theorem extends [6, Theorem 1.3] to the case ≥ 1 s .A proof can be found in Appendix A.
The following abstract lemma provides sufficient conditions to transfer the limsup inequality from approximating elements to the limit element.This result applied to smooth convolutions approximating a given C 1 -curve γ will be the second ingredient in the proof of the limsup inequality, Theorem 4.4 below.

Lemma 4.3 (Limsup inequality by approximation). (X, d) be a metric space and F
We learnt this result from [20, Lemma 1.0.4]; for the convenience of the reader we present its proof in Appendix C.
The proof of the limsup inequality is inspired by Blatt's improvement of Scholtes' Γconvergence result for the Möbius energy [6,Theorem 4.8].
and the limsup inequality follows trivially.From now on let TP q (γ) < ∞.Thus, we have We now consider a sequence of suitably rescaled and reparametrized convolutions of γ and prove the limsup inequality for these convolutions.Applying Lemma 4.3 then yields the limsup inequality for γ.
Step 2: For k ∈ N we set ε k := 1 k .Let γ ε k be the convolution as in (4.1) and L (γ ε k ) the length of γ ε k .We then define γk as the arclength parametrization of the rescalings Lγ ε k /L (γ ε k ).Thus, γk has the same length as γ for every k ∈ N. Furthermore, γk is on [0, L) injective for k sufficiently large, which follows from the bilipschitz property (4.2) of γ together with the C 1 -convergence of the convolutions γ ε k → γ as k → ∞.By omitting finitely many indices we may assume that γk ∈ C ∞ ia (R/LZ, R 3 ) for all k ∈ N.For every k ∈ N there is by Lemma 2.9 some index N 0 (k) ∈ N such that there exist proper γk -interpolating balanced biarc curves βk n parametrized by arclength that interpolate the point-tangent pairs ++ satisfy (see Definitions 2.7 and be the length of βk n , and notice that Theorem 2.11 implies Step 3: For k ∈ N, let ϕ k n be Smutny's reparametrization [30, Appendix A] and define Bk n := βk n • ϕ k n , so that Theorem 2.12 implies γk − Bk n C 1 → 0 for each k ∈ N as n → ∞.
where β k n is the reparametrization of B k n by arclength.
Step 4: We now show that β k n ∈ B n holds if n is sufficiently large, such that the values E n q (β k n ) are finite by definition (1.8) in the introduction.We need to show that the length λ k n,i of the i-th biarc of β k n satisfies (1.7).For that we apply Lemma 2.10 to the length λk n,i of the i-th biarc of βk n .More precisely, we take the limit n → ∞ in the following inequality which holds for each k Combining this with (4.3) we find for each k ∈ N and index which is (1.7) for λ i := λ k i .Thus β k n ∈ B n for all n ≥ N 1 (k).
Step 5: The scaling property (1.9) and the parameter invariance of the discrete tangent-point energies yields so that we obtain by (4.3) and Theorem 3.1 applied to γ := γk and Step 6: In this final step check the assumptions of Lemma 4.3.The space C 1 ia R/LZ, R 3 is a metric space with the metric induced by the C 1 -norm.By the Morrey-Sobolev embedding (see [19,Theorem A.2] in the setting of periodic functions) there exists a constant c E > 0, such that According to Theorem 4.2 applied to = q and s = 1 − 1 q for q > 2 the righthand side converges to 0 as k → ∞.Thus, γk converges in the C 1 -norm to γ, which verifies condition (i) in Lemma 4.3.Furthermore, [34, (4.2) Satz] implies that TP q is continuous on W 2− 1 q ,q ia since q > 2. Thus, we obtain lim k→∞ TP q (γ k ) = TP q (γ), which gives us condition (ii) of Lemma 4.3.Combining (4.5) with (4.8) verifies condition (iii) of Lemma 4.3.Hence, Lemma 4.3 yields the limsup inequality for γ.
Remark 4.5.For the proof of Theorem 1.3 in Section 5 it is important to note that the actual recovery sequence for the limsup inequality in the previous proof is a subsequence of the (doubly subscripted) arclength parametrized biarc curves Convergence of discrete almost minimizers.In this subsection, we prove the convergence of discrete almost minimizers of the discrete tangent-point energies in the metric space defined before.The following lemma of Scholtes [25,Lemma A.1] states the general result.For completeness we present its short proof in Appendix C. Lemma 4.6 (Convergence of minimizers).Let (X, d) be a metric space and F n , F : Proof of Corollary 1.2.The proof follows immediately from Lemma 4.6 with the metric space Y = X = C 1 ia R/LZ, R 3 ∩ K, with metric induced by the C 1 -norm.Notice that the knot class K is stable under C 1 -convergence; see, e.g., [24].Since TP q (γ) < ∞ holds, we obtain γ ∈ W

Γ-convergence to the Ropelength functional
As a first step towards the proof of Theorem 1.3 we show that the continuous tangentpoint energies (TP k ) We follow the proof of [14,Theorem 6.11], where Gilsbach showed Γ-convergence of Integral Menger curvatures towards ropelength.
Furthermore, the continuous tangent-point energy is lower semi-continuous with respect to the C k is also the Γ-limit and we obtain Proof.By Theorem 1.1 we have for any q > 2 as n → ∞.However, in the proof of the limsup inequality in Theorem 4.4 the recovery sequence is a sequence consisting only of biarc curves that are in C 1,1 ia R/Z, R 3 ; see Remark 4.5.Therefore we also have E n q Γ −→ TP q on the space (C 1,1 ia (R/Z, R 3 ), • C 1 ) as n → ∞.Now apply Lemma C.2 in Appendix C for F n := E n q , F := TP q and the continuous and non-decreasing function g : (0, ∞) → R, x → x 1 q to infer (E n q ) Next we compare two different discrete tangent-point energies. Then Proof.We only have to consider the case that γ ∈ B n since otherwise both sides of the inequality are infinite by definition of the discrete energy E n k ; see (1.8) in the introduction.Denote by (([q i , t i ] , [q i+1 , t i+1 ])) i=0,..,n−1 the point-tangent pairs that γ interpolates.For i = j define x i,j := ≥ 0. Then we estimate by means of 3 Be aware of the notation: In [29] the expression R[•] was used for thickness the generalized inequality for finite sums, ( 1 1 q for p ≤ q (here for := n(n − 1), p := k, q := m), and by (1.7) Proof of Theorem 1.3.It suffices to prove the Γ-convergence for L = 1, since then the statement for general L follows from the scaling property and parametrization invariance of the energies involved.Indeed, assume the theorem was proven for L = 1.Now take L = 1 and let Denote by γn the arclength parametrization of γn L .By Lemma B.3 this implies γn → γ in C 1 as n → ∞, where γ is the arclength parametrization of γ L .Together with the fact that the ropelength functional is invariant under reparametrization and scaling, the liminf inequality for L = 1 yields the liminf equality for general L: = lim inf For the limsup inequality let is the arclength parametrization of γ scaled to unit length.Hence, there exists a recovery sequence Define the reparametrization ϕ : [0, L] → [0, 1], x → x L and set γ n (x) := L γn (ϕ(x)) and γ(x) := Lγ(ϕ(x)).Note that γ n is parametrized by arclength and that γ = γ holds.Then γ n → γ = γ in C 1 for n → ∞ by (5.2).Again, by the scaling property of the energies and the invariance under reparametrization we deduce with (5.2) So, it remains to prove the statement of Theorem 1.3 for L = 1, and for that we take a general sequence For k ≤ n we apply Lemma 5.3 to γ := γ n and m := n to find Together with (5.3) and L (γ n ) = 1 for all n ∈ N, this yields (5.4) Now we have Combining this with the pointwise convergence in Lemma 5.1 and (5.4) we arrive at the desired liminf inequality: To verify the limsup inequality let γ ∈ C 1,1 ia R/Z, R 3 , and for n ∈ N set s n,i := i n for i = 0, . . ., n.Then we have |s n,i+1 − s n,i | = 1 n for all i = 0, . . ., n − 1, and therefore a sequence of (c 1 − c 2 )-distributed partitions with c 1 = c 2 = 1; see Defintition 2.8.Now we follow the proof of Theorem 4.4.However, since γ is now a C 1,1 -curve, we do not have to work with convolutions, but can follow the proof for γ directly.By Lemma 2.9 there exists for n sufficiently large a γ-interpolating, proper and balanced biarc curve βn interpolating the point-tangent pairs (([γ(s n,i ), γ (s n,i )], [γ(s n,i+1 ), γ (s n,i+1 )])) i=0,...,n−1 .
Then we obtain by Theorem 2.11 that L ( βn ) → L (γ) = 1 as n → ∞.Let ϕ n be the reparametrization function from [30, Appendix A] and set Bn := βn • ϕ n .Then by Theorem 2.12, we have Bn → γ in C 1 for n → ∞.Setting B n := L ( βn ) −1  Bn we obtain as in the proof of Theorem 4.4 that B n → γ in C 1 for n → ∞.Let β n be the arclength parametrization of B n .By Lemma B.3 we finally arrive at β n → γ in C 1 for n → ∞.The biarc-curves β n are only reparametrized versions of βn rescaled by the factor L ( βn ) −1 so that we can show exactly as in the proof of Theorem 4.4 that β n ∈ B n for n sufficiently large.Moreover, due to the C 1 -convergence towards γ, the β n are also injective for n large enough.Since β n is scaled to unit length and parametrized by arclength, we have β n ∈ C 1,1 ia (R/Z, R 3 ) for n sufficiently large.Set L n := L ( βn ).By the scaling property of the discrete tangent-point energy (1.9) and its parameter invariance we have (5.5) for i, j = 0, . . ., n − 1 with i = j we can write and estimate for sufficiently large n ∈ N for n sufficiently large.Finally, taking the limsup yields the desired limsup inequality Proof of Corollary 1.4.Apply Lemma 4.6 to the metric space Y = X = C 1,1 ia R/Z, R 3 ∩ K with metric induced by the C 1 -norm.Notice as in the proof of Corollary 1.2 that according to [24] the knot class K is stable under C 1 -convergence.Since R(γ) < ∞ holds, we obtain by [16,Lemma 2] For fixed L > 0, s ∈ (0, 1) and ∈ [1, ∞) define the seminorm [f ] s, of an L-periodic locally -integrable function f : R → R n as where |x−y| R/LZ denotes the periodic distance on R defined in (2.3).Then the periodic fractional 4 Sobolev space W 1+s, (R/LZ, R n ) consists of those Sobolev functions f ∈ W 1, (R/LZ, R n ) whose weak derivatives f have finite seminorm [f ] s, .The norm on Proof of Theorem 4.2.The case = 1 s is treated in [6, Theorem 1.3], so we may assume from now on that > 1 s .
So, we need to show the uniform integrability.In the obvious inequality we estimate both summands on the right-hand side separately.This is easy for the second summand.Fix Regarding the first summand in (A.5) we consider the arclength function s It is well-known that the standard convolution γ ε converges in W 1+s, to γ; see, e.g., [12,Lemma 11], which according to Vitali's theorem implies that the A ε (x, y) are uniformly integrable.In particular, for given ε > 0 there exists δ for all ε > 0. (A.11) Since ψ ε is uniformly Lipschitz-continuous for ε ∈ (0, ε 0 ], there exists a δ 3 > 0 such that |E| < δ 3 implies |ψ ε (E)| < δ 2 .Now set δ := min{δ 1 , δ 3 } so that for any set E ⊂ (R/LZ) 2 with |E| < δ we infer by means of (A.5), (A.10), (A.6), and (A.11) Proof.Using the Taylor expansion On the other hand, since γ is injective, we find a constant δ 0 = δ 0 (γ) > 0 such that 3).We use the Taylor expansion (B.2) to estimate and analogously Using the above estimate for the inner product and the Lipschitz estimate |γ(s where ω γ denotes the modulus of continuity of γ and ω γ denotes the modulus of continuity of the tangent γ of γ.
Proof.Without loss of generality, we can assume For every n ∈ N, we then set For the proof of this claim we distinguish three cases.
Case 1: There exists an index N ∈ N such that J n = ∅ is bounded for all n ≥ N .Suppose for contradiction that there exists a constant c > 0 and a subsequence k(n) → ∞ as n → ∞ such that m k(n) ≤ c for all n ≥ N .Thus, m holds for all k ≥ n 0 , which contradicts the first assumption in (C.2).Hence, we conclude that the assumption was wrong and lim n→∞ m n = ∞ must hold in this case.Case 2: J n = ∅ for infinitely many n ∈ N. Then for these n we have, similarly as in (C.2), either F n (x m n ) > F(x m ) + 1 m or x m n ∈ U m for all m.Exactly as in Case 1 both options lead to contradictive statements, which rules out Case 2.
Case 3: There exists a subsequence n l such that J n l is unbounded.By definition of J n l , we get m n l ≥ n l and thus m n l → ∞ as l → ∞.Every other subsequence of m n can then be handled as in case 1.By the subsequence principle we get m n → ∞ as n tends to infinity, which proves the claim.
We now define y n := x mn n for all n ∈ N. By the definition of m n , we get y n ∈ U mn and thus 0 ≤ d(x, y n ) < d(x, x mn ) + 1 mn for every n ∈ N. Applying m n → ∞ as n → ∞ and assumption (i) we then deduce d (x, y n ) → 0 as n → ∞.Furthermore, we estimate Taking the infimum over all y ∈ Y on the right-hand side yields the desired inequality, from which the last two equations follow as claimed if z ∈ Y .
Sending ε to zero, we obtain by continuity of g that g(inf(A)) = inf(g(A)).In the same manner, we prove that sup(g(A)) = g(sup(A)).
Lemma C.2.Let (X, d) be a metric space, F n , F : X → (0, ∞) and Furthermore, let g : (0, ∞) → R be a continuous and monotone increasing map.Then Proof.First, we prove the liminf inequality.Let x ∈ X and x n → x for n → ∞.By assumption, we have F(x) ≤ lim inf n→∞ F n (x n ).This yields with the monotonicity and continuity of g and Lemma C. For the limsup inequality fix x ∈ X.By assumption, there exists a recovery sequence x n → x as n → ∞ such that F(x) ≥ lim sup n→∞ F n (x n ).Then again, we obtain g(F(x)) ≥ g(lim sup

Corollary 1 . 4 (
Discrete almost minimizers approximate ropelength minimizers).Let K be a tame knot class and b n ∈ C * * , or to compute discrete (almost) minimizers of the tangent-point energy.The almost linear energy convergence rate established in Theorem 3.1 in Section 3 is identical with the one in [25, Proposition 3.1] for Scholtes' polygonal Möbius energy, which exceeds the n − 1 4 -convergence rate for the minimal distance approximation of the Möbius energy by Rawdon and Simon [23, Theorem 1].

Figure 1 .
Figure 1.Left and middle: Examples of biarcs, t m is the tangent at the common matching point m.Right: The circles C 0 , C 1 , C + , and C − of Definition 2.2.Images taken from [30, Figures 4.1& 4.3] by courtesy of Jana Smutny.

1 n − 1
Bk n (s) for all s ∈ R. Then B k n has obviously length L. However, B k n is not parametrized by arclength.Nevertheless, by means of (4.3) and (4.4) we findB k n − Bk n C 1 = L L k Bk n C 1 → 0 for each k ∈ N as n → ∞.Consequently, by (4.4), one has γk − B k n C 1 → 0 for each k ∈ N as n → ∞,and therefore by means of Lemma B.3,

− 1
n) −→ 1 as n → ∞. (4.6)Since the image of β k n is just the image of βk n scaled by the factor L L k n we deduce see the choice of the abstract recovery sequence (y n ) n towards the end of the proof of Lemma 4.3 in Appendix C. Proof of Theorem 1.1.According to [8, Definition 1.5] it suffices to verify two fundamental inequalities.Indeed, the liminf inequality is the content of Theorem 4.1, whereas the limsup inequality is established in Theorem 4.4.

m 1 m
or x m k(n) / ∈ U m for all m ≥ c, n ≥ N. (C.2) If x m k(n) / ∈ U m holds for any m ≥ c, we deduce from (C.1) that d(x, x m k(n) ) ≥ d(x, x m )+ 1 m , and therefore, d(x m k(n) , x m ) ≥ d(x, x m k(n) )−d(x, x m ) = 1 m .This contradicts d(x m k(n) , x m ) → 0 as n → ∞ in assumption (iii).Thus, we have x m k(n) ∈ U m for all m ≥ c and consequently F k(n) (x m k(n) ) > F(x m ) + 1 m in (C.2) must hold.By assumption (iii) we havelim sup n→∞ F n (x m n ) ≤ F(x m ), or equivalently inf n∈N sup k≥n F k (x m k ) ≤ F(x m ).By the definition of the infimum, there exists an index n 0 = n 0 (m) ∈ N such thatsup k≥n 0 F k (x m k ) ≤ inf n∈N sup k≥n F k (x m k ) + 1 m ≤ F (x m ) +and thus F k (x m k ) ≤ F(x m )+ 1