Flows on $S^3$ supporting all links as orbits
Author:
Robert W. Ghrist
Journal:
Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 91-97
MSC (1991):
Primary 57M25, 58F22; Secondary 58F25, 34C35
DOI:
https://doi.org/10.1090/S1079-6762-95-02006-3
MathSciNet review:
1350685
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We construct counterexamples to some conjectures of J. Birman and R. F. Williams concerning the knotting and linking of closed orbits of flows on 3-manifolds. By establishing the existence of “universal templates,” we produce examples of flows on $S^3$ containing closed orbits of all knot and link types simultaneously. In particular, the set of closed orbits of any flow transverse to a fibration of the complement of the figure-eight knot in $S^3$ over $S^1$ contains representatives of every (tame) knot and link isotopy class. Our methods involve semiflows on branched 2-manifolds, or templates.
- J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. USA 9 (1923), 93–95.
- D. Asimov and J. Franks, Unremovable closed orbits, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 22–29. MR 730260, DOI https://doi.org/10.1007/BFb0061407
- Joan S. Birman and R. F. Williams, Knotted periodic orbits in dynamical systems. I. Lorenz’s equations, Topology 22 (1983), no. 1, 47–82. MR 682059, DOI https://doi.org/10.1016/0040-9383%2883%2990045-9
- Joan S. Birman and R. F. Williams, Knotted periodic orbits in dynamical system. II. Knot holders for fibered knots, Low-dimensional topology (San Francisco, Calif., 1981) Contemp. Math., vol. 20, Amer. Math. Soc., Providence, RI, 1983, pp. 1–60. MR 718132, DOI https://doi.org/10.1090/conm/020/718132
- Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 82. MR 0375281
- Joan S. Birman, A representation theorem for fibered knots and their monodromy maps, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977) Lecture Notes in Math., vol. 722, Springer, Berlin, 1979, pp. 1–8. MR 547448
- Rufus Bowen, On Axiom A diffeomorphisms, American Mathematical Society, Providence, R.I., 1978. Regional Conference Series in Mathematics, No. 35. MR 0482842
- Leon O. Chua, Motomasa Komuro, and Takashi Matsumoto, The double scroll family. I. Rigorous proof of chaos, IEEE Trans. Circuits and Systems 33 (1986), no. 11, 1072–1097. MR 865057, DOI https://doi.org/10.1109/TCS.1986.1085869
- Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR 568308
- John Franks and R. F. Williams, Entropy and knots, Trans. Amer. Math. Soc. 291 (1985), no. 1, 241–253. MR 797057, DOI https://doi.org/10.1090/S0002-9947-1985-0797057-1
- R. Ghrist, Branched two-manifolds supporting all links, Submitted for publication, December 1994.
- R. Ghrist and P. Holmes, An ODE whose solution contains all knots and links, To appear in Intl. J. Bifurcation and Chaos, 1995.
- Philip Holmes and R. F. Williams, Knotted periodic orbits in suspensions of Smale’s horseshoe: torus knots and bifurcation sequences, Arch. Rational Mech. Anal. 90 (1985), no. 2, 115–194. MR 798342, DOI https://doi.org/10.1007/BF00250717
- Michael C. Sullivan, The prime decomposition of knotted periodic orbits in dynamical systems, J. Knot Theory Ramifications 3 (1994), no. 1, 83–120. MR 1265454, DOI https://doi.org/10.1142/S0218216594000083
- William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI https://doi.org/10.1090/S0273-0979-1982-15003-0
- William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI https://doi.org/10.1090/S0273-0979-1988-15685-6
- J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. USA 9 (1923), 93–95.
- D. Asimov and J. Franks, Unremovable closed orbits, Geometric Dynamics, Lecture Notes in Mathematics 1007 (J. Palis, ed.), Springer-Verlag, 1983.
- J. Birman and R. F. Williams, Knotted periodic orbits in dynamical systems–I : Lorenz’s equations, Topology 22(1) (1983), 47–82.
- ---, Knotted periodic orbits in dynamical systems–II : knot holders for fibered knots, Cont. Math. 20 (1983), 1–60.
- J. S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J., 1974.
- ---, On the construction of fibred knots and their monodromy maps, Topology of Low-Dimensional Manifolds, Lecture Notes in Mathematics 722 (R. Fenn, ed.), Springer-Verlag, 1979.
- R. Bowen, On Axiom A diffeomorphisms, Regional Conference Series in Mathematics 35.
- L. Chua, M. Komuro, and T. Matsumoto, The double scroll family, IEEE Trans. on Circuits and Systems 33 (1986), 1073–1118.; b
- A. Fathi, F. Laudenbach, and V. Poenaru et al., Travaux de Thurston sur les surfaces, Astérique 66-67 (1979), 1–284.
- J. Franks and R. F. Williams, Entropy and knots, Trans. Am. Math. Soc. 291(1) (1985), 241–253.
- R. Ghrist, Branched two-manifolds supporting all links, Submitted for publication, December 1994.
- R. Ghrist and P. Holmes, An ODE whose solution contains all knots and links, To appear in Intl. J. Bifurcation and Chaos, 1995.
- P. J. Holmes and R. F. Williams, Knotted periodic orbits in suspensions of Smale’s horseshoe: torus knots and bifurcation sequences, Archive for Rational Mech. and Anal. 90(2) (1985), 115 –193.
- M. C. Sullivan, The prime decomposition of knotted periodic orbits in dynamical systems, J. Knot Thy. and Ram. 3(1) (1994), 83–120.
- W. P. Thurston, Three dimensional manifolds, Kleinian groups, and hyperbolic geometry, Bull. Am. Math. Soc. 6(3) (1982), 357–381.
- ---, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Am. Math. Soc. 19(2) (1988), 417–431.
Similar Articles
Retrieve articles in Electronic Research Announcements of the American Mathematical Society
with MSC (1991):
57M25,
58F22,
58F25,
34C35
Retrieve articles in all journals
with MSC (1991):
57M25,
58F22,
58F25,
34C35
Additional Information
Robert W. Ghrist
Affiliation:
Center for Applied Mathematics, Cornell University, Ithaca NY, 14853
Address at time of publication:
Program in Applied and Computational Mathematics, Princeton University, Princeton NJ, 08544-1000; Institute for Advanced Study, Princeton NJ, 08540
Email:
rwghrist@math.princeton.edu; robg@math.ias.edu
Keywords:
Knots,
links,
branched 2-manifolds,
flows.
Received by editor(s):
June 16, 1995
Additional Notes:
The author was supported in part by an NSF Graduate Research Fellowship.
The author wishes to thank Philip Holmes for his encouragement and support.
Article copyright:
© Copyright 1995
American Mathematical Society
