On the distortion of knots on embedded surfaces. (English) Zbl 1227.57013

Summary: Our main result is a nontrivial lower bound for the distortion of some specific knots. In particular, we show that the distortion of the torus knot \(T_{p,q}\) satisfies \(\delta(T_{p,q}) \geq \frac 1{160}\min(p,q)\). This answers a 1983 question of Gromov.


57M25 Knots and links in the \(3\)-sphere (MSC2010)
53A04 Curves in Euclidean and related spaces


knot distortion
Full Text: DOI arXiv


[1] R. H. Bing, ”An alternative proof that \(3\)-manifolds can be triangulated,” Ann. of Math., vol. 69, pp. 37-65, 1959. · Zbl 0106.16604
[2] E. Denne and J. M. Sullivan, ”The distortion of a knotted curve,” Proc. Amer. Math. Soc., vol. 137, iss. 3, pp. 1139-1148, 2009. · Zbl 1179.53003
[3] R. H. Fox, ”A remarkable simple closed curve,” Ann. of Math., vol. 50, pp. 264-265, 1949. · Zbl 0033.13603
[4] M. H. Freedman, Z. He, and Z. Wang, ”Möbius energy of knots and unknots,” Ann. of Math., vol. 139, iss. 1, pp. 1-50, 1994. · Zbl 0817.57011
[5] M. Gromov, ”Homotopical effects of dilatation,” J. Differential Geom., vol. 13, iss. 3, pp. 303-310, 1978. · Zbl 0427.58010
[6] M. Gromov, ”Filling Riemannian manifolds,” J. Differential Geom., vol. 18, iss. 1, pp. 1-147, 1983. · Zbl 0515.53037
[7] A. J. S. Hamilton, ”The triangulation of \(3\)-manifolds,” Quart. J. Math. Oxford Ser., vol. 27, iss. 105, pp. 63-70, 1976. · Zbl 0318.57003
[8] R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, with notes by J. Milnor and M. Atiyah, Princeton, N.J.: Princeton Univ. Press, 1977, vol. 88. · Zbl 0361.57004
[9] E. E. Moise, ”Affine structures in \(3\)-manifolds. V. The triangulation theorem and Hauptvermutung,” Ann. of Math., vol. 56, pp. 96-114, 1952. · Zbl 0048.17102
[10] C. A. S. Mullikin, ”A class of curves in every knot type where chords of high distortion are common,” Topology Appl., vol. 154, iss. 14, pp. 2697-2708, 2007. · Zbl 1139.57006
[11] J. Munkres, ”Obstructions to the smoothing of piecewise-differentiable homeomorphisms,” Ann. of Math., vol. 72, pp. 521-554, 1960. · Zbl 0108.18101
[12] J. O’Hara, ”Family of energy functionals of knots,” Topology Appl., vol. 48, iss. 2, pp. 147-161, 1992. · Zbl 0769.57006
[13] J. O’Hara, ”Energy functionals of knots. II,” Topology Appl., vol. 56, iss. 1, pp. 45-61, 1994. · Zbl 0996.57503
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