Abstract
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.

[bing] R. H. Bing, "An alternative proof that $3$manifolds can be triangulated," Ann. of Math., vol. 69, pp. 3765, 1959.
@article {bing, MRKEY = {0100841},
AUTHOR = {Bing, R. H.},
TITLE = {An alternative proof that {$3$}manifolds can be triangulated},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {69},
YEAR = {1959},
PAGES = {3765},
ISSN = {0003486X},
MRCLASS = {55.00},
MRNUMBER = {0100841},
MRREVIEWER = {S. S. Cairns},
DOI = {10.2307/1970092},
ZBLNUMBER = {0106.16604},
} 
[dennesullivan] E. Denne and J. M. Sullivan, "The distortion of a knotted curve," Proc. Amer. Math. Soc., vol. 137, iss. 3, pp. 11391148, 2009.
@article {dennesullivan, MRKEY = {2457456},
AUTHOR = {Denne, Elizabeth and Sullivan, John M.},
TITLE = {The distortion of a knotted curve},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {137},
YEAR = {2009},
NUMBER = {3},
PAGES = {11391148},
ISSN = {00029939},
CODEN = {PAMYAR},
MRCLASS = {57M25 (53A04 58E10)},
MRNUMBER = {2457456},
MRREVIEWER = {Dave Auckly},
DOI = {10.1090/S0002993908097050},
ZBLNUMBER = {1179.53003},
} 
@article {foxwild, MRKEY = {0030745},
AUTHOR = {Fox, Ralph H.},
TITLE = {A remarkable simple closed curve},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {50},
YEAR = {1949},
PAGES = {264265},
ISSN = {0003486X},
MRCLASS = {56.0X},
MRNUMBER = {0030745},
MRREVIEWER = {S. Eilenberg},
DOI = {10.2307/1969450},
ZBLNUMBER = {0033.13603},
} 
[freedman] M. H. Freedman, Z. He, and Z. Wang, "Möbius energy of knots and unknots," Ann. of Math., vol. 139, iss. 1, pp. 150, 1994.
@article {freedman, MRKEY = {1259363},
AUTHOR = {Freedman, Michael H. and He, ZhengXu and Wang, Zhenghan},
TITLE = {Möbius energy of knots and unknots},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {139},
YEAR = {1994},
NUMBER = {1},
PAGES = {150},
ISSN = {0003486X},
CODEN = {ANMAAH},
MRCLASS = {58E10 (57M25)},
MRNUMBER = {1259363},
MRREVIEWER = {Anders Linn{é}r},
DOI = {10.2307/2946626},
ZBLNUMBER = {0817.57011},
} 
[gromov1] M. Gromov, "Homotopical effects of dilatation," J. Differential Geom., vol. 13, iss. 3, pp. 303310, 1978.
@article {gromov1, MRKEY = {0551562},
AUTHOR = {Gromov, Mikhael},
TITLE = {Homotopical effects of dilatation},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {13},
YEAR = {1978},
NUMBER = {3},
PAGES = {303310},
ISSN = {0022040X},
CODEN = {JDGEAS},
MRCLASS = {58D15 (58E10 58E15)},
MRNUMBER = {0551562},
MRREVIEWER = {Y. Mut{ô}},
ZBLNUMBER = {0427.58010},
URL = {http://projecteuclid.org/euclid.jdg/1214434601},
} 
[gromov2] M. Gromov, "Filling Riemannian manifolds," J. Differential Geom., vol. 18, iss. 1, pp. 1147, 1983.
@article {gromov2, MRKEY = {0697984},
AUTHOR = {Gromov, Mikhael},
TITLE = {Filling {R}iemannian manifolds},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {18},
YEAR = {1983},
NUMBER = {1},
PAGES = {1147},
ISSN = {0022040X},
CODEN = {JDGEAS},
MRCLASS = {53C20 (53C21 57R99)},
MRNUMBER = {0697984},
MRREVIEWER = {Yu. Burago},
ZBLNUMBER = {0515.53037},
URL = {http://projecteuclid.org/euclid.jdg/1214509283},
} 
[hamilton] A. J. S. Hamilton, "The triangulation of $3$manifolds," Quart. J. Math. Oxford Ser., vol. 27, iss. 105, pp. 6370, 1976.
@article {hamilton, MRKEY = {0407848},
AUTHOR = {Hamilton, A. J. S.},
TITLE = {The triangulation of {$3$}manifolds},
JOURNAL = {Quart. J. Math. Oxford Ser.},
FJOURNAL = {The Quarterly Journal of Mathematics. Oxford. Second Series},
VOLUME = {27},
YEAR = {1976},
NUMBER = {105},
PAGES = {6370},
ISSN = {00335606},
MRCLASS = {57C15},
MRNUMBER = {0407848},
MRREVIEWER = {Mario Pezzana},
ZBLNUMBER = {0318.57003},
DOI = {10.1093/qmath/27.1.63},
} 
[kirsie] 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.
@book{kirsie, MRKEY = {0645390},
AUTHOR={Kirby, Robion C. and Siebenmann, L. C.},
TITLE={Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, {\rm with notes by J. Milnor and M. Atiyah}},
PUBLISHER={Princeton Univ. Press},
ADDRESS={Princeton, N.J.},
YEAR={1977},
SERIES={Ann. of Math. Studies},
VOLUME={88},
MRNUMBER = {0645390},
ZBLNUMBER = {0361.57004},
} 
[moise] E. E. Moise, "Affine structures in $3$manifolds. V. The triangulation theorem and Hauptvermutung," Ann. of Math., vol. 56, pp. 96114, 1952.
@article {moise, MRKEY = {0048805},
AUTHOR = {Moise, Edwin E.},
TITLE = {Affine structures in {$3$}manifolds. {V}. {T}he triangulation theorem and {H}auptvermutung},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {56},
YEAR = {1952},
PAGES = {96114},
ISSN = {0003486X},
MRCLASS = {56.0X},
MRNUMBER = {0048805},
MRREVIEWER = {S. S. Cairns},
DOI = {10.2307/1969769},
ZBLNUMBER = {0048.17102},
} 
[mullikin] 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. 26972708, 2007.
@article {mullikin, MRKEY = {2340952},
AUTHOR = {Mullikin, Chad A. S.},
TITLE = {A class of curves in every knot type where chords of high distortion are common},
JOURNAL = {Topology Appl.},
FJOURNAL = {Topology and its Applications},
VOLUME = {154},
YEAR = {2007},
NUMBER = {14},
PAGES = {26972708},
ISSN = {01668641},
CODEN = {TIAPD9},
MRCLASS = {58E10 (57M25)},
MRNUMBER = {2340952},
MRREVIEWER = {Anders Linn{é}r},
DOI = {10.1016/j.topol.2007.05.003},
ZBLNUMBER = {1139.57006},
} 
[munkres] J. Munkres, "Obstructions to the smoothing of piecewisedifferentiable homeomorphisms," Ann. of Math., vol. 72, pp. 521554, 1960.
@article {munkres, MRKEY = {0121804},
AUTHOR = {Munkres, James},
TITLE = {Obstructions to the smoothing of piecewisedifferentiable homeomorphisms},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {72},
YEAR = {1960},
PAGES = {521554},
ISSN = {0003486X},
MRCLASS = {57.00},
MRNUMBER = {0121804},
MRREVIEWER = {P. Dedecker},
DOI = {10.2307/1970228},
ZBLNUMBER = {0108.18101},
} 
[ohara1] J. O’Hara, "Family of energy functionals of knots," Topology Appl., vol. 48, iss. 2, pp. 147161, 1992.
@article {ohara1, MRKEY = {1195506},
AUTHOR = {O'Hara, Jun},
TITLE = {Family of energy functionals of knots},
JOURNAL = {Topology Appl.},
FJOURNAL = {Topology and its Applications},
VOLUME = {48},
YEAR = {1992},
NUMBER = {2},
PAGES = {147161},
ISSN = {01668641},
CODEN = {TIAPD9},
MRCLASS = {58E20 (57M25 58E10)},
MRNUMBER = {1195506},
MRREVIEWER = {Caio J. C. Negreiros},
DOI = {10.1016/01668641(92)90023S},
ZBLNUMBER = {0769.57006},
} 
[ohara2] J. O’Hara, "Energy functionals of knots. II," Topology Appl., vol. 56, iss. 1, pp. 4561, 1994.
@article {ohara2, MRKEY = {1261169},
AUTHOR = {O'Hara, Jun},
TITLE = {Energy functionals of knots. {II}},
JOURNAL = {Topology Appl.},
FJOURNAL = {Topology and its Applications},
VOLUME = {56},
YEAR = {1994},
NUMBER = {1},
PAGES = {4561},
ISSN = {01668641},
CODEN = {TIAPD9},
MRCLASS = {58D10 (49Q10 57M25 58E10)},
MRNUMBER = {1261169},
MRREVIEWER = {Anders Linn{é}r},
DOI = {10.1016/01668641(94)901082},
ZBLNUMBER = {0996.57503},
}