The universal templates of Ghrist

Author:
R. F. Williams

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

MSC (1991):
Primary 57-XX

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]**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**, 10.1016/0040-9383(83)90045-9**[B-W2]**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**, 10.1090/conm/020/718132**[F]**John M. Franks,*Knots, links and symbolic dynamics*, Ann. of Math. (2)**113**(1981), no. 3, 529–552. MR**621015**, 10.2307/2006996**[F-W]**John Franks and R. F. Williams,*Entropy and knots*, Trans. Amer. Math. Soc.**291**(1985), no. 1, 241–253. MR**797057**, 10.1090/S0002-9947-1985-0797057-1**[G]**Robert W. Ghrist,*Branched two-manifolds supporting all links*, Topology**36**(1997), no. 2, 423–448. MR**1415597**, 10.1016/0040-9383(96)00006-7**[G-H]**Robert Ghrist and Philip Holmes,*Knots and orbit genealogies in three-dimensional flows*, Bifurcations and periodic orbits of vector fields (Montreal, PQ, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 408, Kluwer Acad. Publ., Dordrecht, 1993, pp. 185–239. MR**1258520****[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]**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**, 10.1007/BF00250717**[S1]**Mike Sullivan,*Prime decomposition of knots in Lorenz-like templates*, J. Knot Theory Ramifications**2**(1993), no. 4, 453–462. MR**1247579**, 10.1142/S021821659300026X**[S2]**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**, 10.1142/S0218216594000083**[W1]**R. F. Williams,*Expanding attractors*, Colloque de Topologie Différentielle (Mont-Aigoual, 1969) Université de Montpellier, Montpellier, 1969, pp. 79–89. MR**0287581****[W2]**R. F. Williams,*Expanding attractors*, Inst. Hautes Études Sci. Publ. Math.**43**(1974), 169–203. MR**0348794****[W3]**R. F. Williams,*The structure of Lorenz attractors*, Inst. Hautes Études Sci. Publ. Math.**50**(1979), 73–99. MR**556583****[W4]**R. F. Williams,*Lorenz knots are prime*, Ergodic Theory Dynam. Systems**4**(1984), no. 1, 147–163. MR**758900**, 10.1017/S0143385700002339**[W5]**R. F. Williams,*The structure of Lorenz attractors*, Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977) Springer, Berlin, 1977, pp. 94–112. Lecture Notes in Math., Vol. 615. MR**0461581**

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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:
http://dx.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