Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family \(E^{(\alpha )}, \alpha \in [2, 3)\). (English) Zbl 1248.42009
Knot energy functionals were originally introduced to simplify knots experimentally. Further aims regarding energies were in studying critical knots (perfect forms) for energy functionals and appropriate gradient and descent flows taking an arbitrary knot to a perfect form. This paper is dedicated to smoothness and variational aspects of J. O’Hara’s knot functionals \(E^{(\alpha)}\), \(\alpha\in [2, 3)\). The author derives continuity of \(E^{(\alpha)}\) on injective and regular \(H^2\) knots, establishes Fréchet differentiability of \(E^{(\alpha)}\) and then states several first variation formulae. He proves \(C^\infty\)-smoothness of critical points of the appropriately rescaled functionals equation image by means of fractional Sobolev spaces on a periodic interval and bilinear Fourier multipliers.
Reviewer: Oleg Karpenkov (Graz)
MSC:
| 42A45 | Multipliers in one variable harmonic analysis |
| 53A04 | Curves in Euclidean and related spaces |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
References:
| [1] | Abrams, Circles minimize most knot energies, Topology 42 (2) pp 381– (2003) · Zbl 1030.57006 · doi:10.1016/S0040-9383(02)00016-2 |
| [2] | Alt, Energetics and dynamics of global integrals modeling interaction between stiff filaments, J. Math. Biol. 59 (3) pp 377– (2009) · Zbl 1311.92030 · doi:10.1007/s00285-008-0227-6 |
| [3] | T. Ashton J. Cantarella M. Piatek E. Rawdon Self-contact sets for 50 tightly knotted and linked tubes 2005 http://arxiv.org/abs/math/0508248 |
| [4] | Auckly, Geometric topology (Athens, GA, 1993), AMS/IP Stud. Adv. Math. Vol. 2 pp 235– (1997) |
| [5] | Blatt, The gradient flow of the Möbius energy near local minima, Preprint (2009) |
| [6] | S. Blatt Boundedness and regularizing effects of O’Hara’s knot energies 2011 · Zbl 1238.57007 |
| [7] | S. Blatt The gradient flow of O’Hara’s knot energies 2011 · Zbl 1398.53071 |
| [8] | Blatt, Does finite knot energy lead to differentiability, J. Knot Theory Ramifications 17 (10) pp 1281– (2008) · Zbl 1177.53010 · doi:10.1142/S0218216508006622 |
| [9] | S. Blatt Ph. Reiter Stationary points of O’Hara’s knot energies 2011 |
| [10] | Burde, Knots, de Gruyter Studies in Mathematics second ed. Vol. 5 (2003) |
| [11] | Cantarella, Criticality for the Gehring link problem, Geom. Topol. 10 pp 2055– (2006) · Zbl 1129.57006 · doi:10.2140/gt.2006.10.2055 |
| [12] | Cantarella, Tight knot values deviate from linear relations, Nature 392 pp 237– (1998) · doi:10.1038/32558 |
| [13] | Cantarella, On the minimum ropelength of knots and links, Invent. Math. 150 (2) pp 257– (2002) · Zbl 1036.57001 · doi:10.1007/s00222-002-0234-y |
| [14] | Cantarella, Visualizing the tightening of knots, in VIS ’05: Proceedings of the conference on Visualization ’05, IEEE Computer Society pp 575– (2005) |
| [15] | Carlen, Biarcs, global radius of curvature, and the computation of ideal knot shapes, in: Physical and Numerical Models in Knot Theory, Ser. Knots Everything Vol. 36 pp 75– (2005) · Zbl 1095.57004 |
| [16] | Coifman, Au delà des Opérateurs Pseudo différentiels, in: Astérisque Vol. 57 (1978) |
| [17] | Crowell, Graduate Texts in Mathematics Vol. 57 (1977) |
| [18] | Evans, Graduate Studies in Mathematics Vol. 19 (1998) |
| [19] | Freedman, Möbius energy of knots and unknots, Ann. Math. (2) 139 (1) pp 1– (1994) · Zbl 0817.57011 · doi:10.2307/2946626 |
| [20] | Fukuhara, A Fête of Topology pp 443– (1988) · doi:10.1016/B978-0-12-480440-1.50025-3 |
| [21] | H. Gerlach Ideal Knots and other Packing Problems of Tubes 2010 |
| [22] | H. Gerlach J. H. Maddocks \documentclass{article}\usepackage{amssymb, mathrsfs}\begin{document}\pagestyle{empty}\({\protect \mathbb {S}}^3\)\end{document} 2007 |
| [23] | Gonzalez, Existence of ideal knots, J. Knot Theory Ramifications 12 (1) pp 123– (2003) · Zbl 1028.57008 · doi:10.1142/S0218216503002354 |
| [24] | Gonzalez, Global curvature, thickness, and the ideal shapes of knots, Proc. Natl. Acad. Sci. USA 96 (9) pp 4769– (1999) · Zbl 1057.57500 · doi:10.1073/pnas.96.9.4769 |
| [25] | Gonzalez, Global curvature and self-contact of nonlinearly elastic curves and rods, Calc. Var. Partial Differ. Equ. 14 (1) pp 29– (2002) · Zbl 1006.49001 · doi:10.1007/s005260100089 |
| [26] | O. Gonzalez J. H. Maddocks J. Smutny \documentclass{article}\usepackage{amssymb, mathrsfs}\begin{document}\pagestyle{empty}\(\mathbb R^3\)\end{document} 2002 195 215 |
| [27] | Grafakos, Graduate Texts in Mathematics second ed. Vol. 249 (2008) |
| [28] | Grafakos, Graduate Texts in Mathematics second ed. Vol. 250 (2009) |
| [29] | He, A formula for the non-integer powers of the Laplacian, Acta Math. Sin. Engl. Ser. 15 (1) pp 21– (1999) · Zbl 0939.46022 · doi:10.1007/s10114-999-0058-4 |
| [30] | He, The Euler-Lagrange equation and heat flow for the Möbius energy, Commun. Pure Appl. Math. 53 (4) pp 399– (2000) · Zbl 1042.53043 · doi:10.1002/(SICI)1097-0312(200004)53:4<399::AID-CPA1>3.0.CO;2-D |
| [31] | Kato, Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math. 41 (7) pp 891– (1988) · Zbl 0671.35066 · doi:10.1002/cpa.3160410704 |
| [32] | Kusner, Ideal Knots, Ser. Knots Everything, Vol. 19 pp 315– (1998) |
| [33] | Meyer, Cambridge Studies in Advanced Mathematics Vol. 48 (1997) |
| [34] | O’Hara, Energy of a knot, Topology 30 (2) pp 241– (1991) · Zbl 0733.57005 · doi:10.1016/0040-9383(91)90010-2 |
| [35] | O’Hara, Family of energy functionals of knots, Topology Appl. 48 (2) pp 147– (1992) · Zbl 0769.57006 · doi:10.1016/0166-8641(92)90023-S |
| [36] | O’Hara, Energy functionals of knots. II, Topology Appl. 56 (1) pp 45– (1994) · Zbl 0996.57503 · doi:10.1016/0166-8641(94)90108-2 |
| [37] | O’Hara, Knots and Everything Vol. 33 (2003) |
| [38] | Palais, Foundations of global non-linear analysis (1968) |
| [39] | Ph. Reiter All curves in a C 1 -neighbourhood of a given embedded curve are isotopic 2005 http://www.instmath.rwth-aachen.de/Preprints/reiter20051017.pdf |
| [40] | Ph. Reiter Repulsive Knot Energies and Pseudodifferential Calculus 2009 http://darwin. bth.rwth-aachen.de/opus3/volltexte/2009/2848 |
| [41] | Rudin, Real and Complex Analysis (1987) |
| [42] | Schuricht, Euler-Lagrange equations for nonlinearly elastic rods with self-contact, Arch. Ration. Mech. Anal. 168 (1) pp 35– (2003) · Zbl 1030.74029 · doi:10.1007/s00205-003-0253-x |
| [43] | Schuricht, Characterization of ideal knots, Calc. Var. Partial Differ. Equ. 19 (3) pp 281– (2004) · Zbl 1352.58004 · doi:10.1007/s00526-003-0216-y |
| [44] | J. Smutny Global Radii of Curvature, and the Biarc Approximation of Space Curves 2004 |
| [45] | Strzelecki, A geometric curvature double integral of Menger type for space curves, Ann. Acad. Sci. Fenn. Math. 34 (1) pp 195– (2009) · Zbl 1188.49016 |
| [46] | Strzelecki, Regularizing and self-avoidance effects of integral menger curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) IX (1) pp 145– (2010) · Zbl 1193.28007 |
| [47] | Strzelecki, Physical and Numerical Models in Knot Theory, Ser. Knots Everything Vol. 36 pp 547– (2005) · doi:10.1142/9789812703460_0027 |
| [48] | Strzelecki, Global curvature for surfaces and area minimization under a thickness constraint, Calc. Var. Partial Differ. Equ. 25 (4) pp 431– (2006) · Zbl 1096.53003 · doi:10.1007/s00526-005-0334-9 |
| [49] | Strzelecki, On rectifiable curves with Lp-bounds on global curvature: self-avoidance, regularity, and minimizing knots, Math. Z. 257 (1) pp 107– (2007) · Zbl 1354.49028 · doi:10.1007/s00209-007-0117-4 |
| [50] | Strzelecki, Integral Menger curvature for surfaces, Adv. Math. 226 (3) pp 2233– (2011) · Zbl 1211.28003 · doi:10.1016/j.aim.2010.09.016 |
| [51] | Taylor, Partial Differential Equations. I, in: Applied Mathematical Sciences Vol. 115 (1996) |
| [52] | Taylor, Applied Mathematical Sciences Vol. 117 (1997) |
| [53] | Ziemer, Graduate Texts in Mathematics Vol. 120 (1989) |
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