The universal templates of Ghrist

Author:
R. F. Williams

Journal:
Bull. Amer. Math. Soc. **35** (1998), 145-156

MSC (1991):
Primary 57-XX

DOI:
https://doi.org/10.1090/S0273-0979-98-00744-7

MathSciNet review:
1602073

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Abstract | References | Similar Articles | Additional Information

Abstract: This is a report on recent work of Robert Ghrist in which he shows that universal templates exist. Put another way, there are many structurally stable flows in the 3-sphere, each of which has periodic orbits representing every knot type. This answers a question raised originally by Mo Hirsch and popularized by the contrary conjecture by Joan Birman and the present author.

**[B-W1]**Birman, J. and Williams, R.F.,*Knotted periodic orbits in dynamical systems I: Lorenz's equations*, Topology**22**((1)) (1983), 47-82. MR**84k:58138****[B-W2]**-,*Knotted periodic orbits in dynamical systems, II: Knot holders for fibered knots*, Contemporary Mathematics**20**(1983), Amer. Math. Soc., Providence, RI, pp. 1-60. MR**86a:58084****[F]**Franks, J.,*Knots, links, and symbolic dynamics*, Annals of Math.**113**(1981), 529-552. MR**83h:58074****[F-W]**Franks, J. and Williams, R.F.,*Entropy and knots*, Transactions American Mathematical Society**291**((1)) (1985), 241-253. MR**86k:58105****[G]**Ghrist, R.,*Branched two-manifolds supporting all links*, Topology**36(2)**(1997), 423-448. MR**98b:57009****[G-H]**Ghrist, R. and Holmes, P.,*Knots and orbit genealogies in three dimensional flows*, Bifurcations and periodic orbits of vector Fields, NATO ASI series C, vol. 408, Kluwer Academic Press, 1993, pp. 185-239. MR**95g:58192****[G-H-S]**R. Ghrist, P. Holmes, and M. Sullivan,*Knots and links in three-dimensional flows*, Lecture notes in Mathematics, vol. 1654, Springer, 1997. CMP**98:04****[H-W]**Holmes, P. and Williams, R.F.,*Knotted periodic orbits in suspensions of Smale's horseshoe: torus knots and bifurcation sequences*, Archive for Rational Mechanics and Analysis**90**((2)) (1985), 115-193. MR**87h:58142****[S1]**Sullivan, M.C.,*Prime decomposition of knots in Lorenz-like templates*, J. Knot Theory and Its Ramifications**2**((4)) (1993), 453-462. MR**94k:57017****[S2]**-,*The prime decomposition of knotted periodic orbits in dynamical systems*, J. Knot Theory and Its Ramifications**3**((1)) (1994), 83-120. MR**95b:57012****[W1]**Williams, R. F.,*Expanding attractors*, Colloque de Topologie Differentielle, Mont-Aiguall 1969, Universite de Montpellier, 1969, pp. 79-89. MR**44:4784****[W2]**-,*Expanding attractors*, Publicationes des Institute des Hautes Etudes Scientifique**43**(1974), 169-203. MR**50:1289****[W3]**-,*The structure of Lorenz attractors*, Publications Mathematique I.H.E.S.**50**(1979), 321-347. MR**82b:58055b****[W4]**-,*Lorenz knots are prime*, Ergodic theory and dynamical systems**4**(1983), 147-163. MR**86c:58103****[W5]**-,*The structure of Lorenz attractors*, Turbulence Seminar, Springer Lecture Notes, vol. 615, 1977, pp. 94-112. MR**57:1566**

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Additional Information

**R. F. Williams**

Affiliation:
Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712-1082

Address at time of publication:
The Institute for Advanced Study, Princeton, New Jersey 08540

Email:
bob@math.utexas.edu, bobwill@math.ias.edu

DOI:
https://doi.org/10.1090/S0273-0979-98-00744-7

Received by editor(s):
June 3, 1997

Received by editor(s) in revised form:
August 10, 1997, and January 21, 1998

Additional Notes:
Supported in part by a grant from the National Science Foundation.

Thanks to Robert Ghrist for his help, in particular for help in drawing the figures.

We thank the Mathematics Department of Montana State University for their hospitality.

Article copyright:
© Copyright 1998
American Mathematical Society