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Contact topology and hydrodynamics III: knotted orbits
Author(s):
John
Etnyre;
Robert
Ghrist
Journal:
Trans. Amer. Math. Soc.
352
(2000),
5781-5794.
MSC (2000):
Primary 57M25, 37J55;
Secondary 37C27, 76B47
Posted:
August 8, 2000
MathSciNet review:
1781279
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Abstract:
We employ the relationship between contact structures and Beltrami fields derived in part I of this series to construct a steady nonsingular solution to the Euler equations on a Riemannian  whose flowlines trace out closed curves of all possible knot and link types. Using careful contact-topological controls, we can make such vector fields real-analytic and transverse to the tight contact structure on  . Sufficient review of concepts is included to make this paper independent of the previous works in this series.
References:
-
- [Aeb94]
- B. Aebischer et al.
Symplectic Geometry.
Number 124 in Progress in Math. Birkhaüser, Berlin, 1994. MR 96a:58082 - [AK98]
- V. I. Arnold and B. Khesin.
Topological Methods in Hydrodynamics.
Springer-Verlag, Berlin, Heidelberg, New York, 1998. MR 99b:58002 - [Ben83]
- D. Bennequin.
Entrelacements et équations de Pfaff.
Asterisque, 107-108:87-161, 1983. MR 86e:58070 - [BW83a]
- J. Birman and R. Williams.
Knotted periodic orbits in dynamical systems-I : Lorenz's equations.
Topology, 22(1):47-82, 1983. MR 84k:58138 - [BW83b]
- J. Birman and R. Williams.
Knotted periodic orbits in dynamical systems-II : knot holders for fibered knots.
Cont. Math., 20:1-60, 1983. MR 86a:58084 - [Col99]
- V. Colin.
Recollement de variétés de contact tendues. Bull. Soc. Math. France 127:43-69, 1999. CMP 99:15 - [DFG
 +86] - T. Dombre, U. Frisch, J. Greene, M. Hénon, A. Mehr, and A. Soward.
Chaotic streamlines in the ABC flows.
J. Fluid Mech., 167:353-391, 1986. MR 88f:76012 - [EG98]
- J. Etnyre and R. Ghrist.
Contact topology and hydrodynamics I: Beltrami fields and the Seifert conjecture. Nonlinearity, 13:441-458, 2000. CMP 2000:09 - [EG99]
- J. Etnyre and R. Ghrist.
Stratified integrals and unknots in inviscid flows.
Cont. Math., 246:99-111, 1999. CMP 2000:07 - [Eli89]
- Y. Eliashberg.
Classification of overtwisted contact structures on 3-manifolds.
Invent. Math., 98:623-637, 1989. MR 90k:53064 - [Eli92]
- Y. Eliashberg.
Contact 3-manifolds twenty years since J. Martinet's work.
Ann. Inst. Fourier, Grenoble, 42(1-2):165-192, 1992. MR 93k:57029 - [Eli93]
- Y. Eliashberg.
Legendrian and transversal knots in tight contact -manifolds.
In Topological methods in modern mathematics (Stony Brook, NY, 1991), pages 171-193. Publish or Perish, Houston, TX, 1993. MR 94e:57005 - [Gau85]
- J.-L. Gautero.
Chaos lagrangien pour une classe d'écoulements de Beltrami.
C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 301(15):1095-1098, 1985. MR 87c:58072 - [GH93]
- R. Ghrist and P. Holmes.
Knots and orbit genealogies in three dimensional flows.
In Bifurcations and Periodic Orbits of Vector Fields, pages 185-239. NATO ASI series C volume 408, Kluwer Academic Press, 1993. MR 95g:58192 - [GH96]
- R. Ghrist and P. Holmes.
An ODE whose solutions contain all knots and links.
Intl. J. Bifurcation and Chaos, 6(5):779-800, 1996. MR 97j:58127 - [Ghr97]
- R. Ghrist.
Branched two-manifolds supporting all links.
Topology, 36(2):423-447, 1997. MR 98b:57009 - [GHS97]
- R. Ghrist, P. Holmes, and M. Sullivan.
Knots and Links in Three-Dimensional Flows, volume 1654 of Springer Lecture Notes in Mathematics.
Springer-Verlag, Berlin, Heidelberg, New York, 1997. MR 98i:58199 - [GW79]
- J. Guckenheimer and R. Williams.
Structural stability of Lorenz attractors.
Inst. Hautes Études Sci. Publ. Math., 50:59-72, 1979. MR 82b:58055a - [Hof93]
- H. Hofer.
Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three.
Invent. Math., 114:515-563, 1993. MR 94j:58064 - [Hol86]
- P. Holmes.
Knotted periodic orbits in suspensions of Smale's horseshoe: period mutiplying and cabled knots.
Physica D, 21:7-41, 1986. MR 88b:58112 - [Hol87]
- P. Holmes.
Knotted periodic orbits in suspensions of annulus maps.
Proc. Roy. London Soc. A, 411:351-378, 1987. MR 88g:58160 - [HW85]
- P. 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):115 -193, 1985. MR 87h:58142 - [HWZ96]
- H. Hofer, K. Wysocki, and E. Zehnder.
Unknotted periodic orbits for Reeb flows on the three-sphere.
Topol. Methods Nonlinear Anal., 7(2):219-244, 1996. MR 98h:58155 - [HZD98]
- D.-B. Huang, X.-H. Zhao, and H.-H. Dai.
Invariant tori and chaotic streamlines in the ABC flow.
Phys. Lett. A, 237(3):136-140, 1998. MR 98m:76088 - [ML98]
- S. Makar-Limanov.
Tight contact structures on solid tori.
Trans. Am. Math. Soc., 350:1013-1044, 1998. MR 98e:58046 - [Mof85]
- H. Moffatt.
Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology: part I.
J. Fluid Mech., 159:359-378, 1985. MR 87c:76132 - [Mof86]
- H. Moffatt.
Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology: part II.
J. Fluid Mech., 166:359-378, 1986. - [MS95]
- D. McDuff and D. Salamon.
Introduction to Symplectic Topology.
Oxford University Press, New York, 1995. MR 97b:58062 - [Tho69]
- W. Thomson.
On vortex motion.
Trans. R. Soc. Edin., 25:217-260, 1869. - [Wil77]
- R. Williams.
The structure of Lorenz attractors.
In A. Chorin, J. Marsden, and S. Smale, editors, Turbulence Seminar, Berkeley 1976/77, volume 615 of Springer Lecture Notes in Mathematics, pages 94-116, 1977. MR 57:1566 - [Wil98]
- R. Williams.
The universal templates of Ghrist.
Bull. Am. Math. Soc., 35(2):145-156, 1998. CMP 98:12 - [ZKLH93]
- X.-H. Zhao, K.-H. Kwek, J.-B. Li, and K.-L. Huang.
Chaotic and resonant streamlines in the ABC flow.
SIAM J. Appl. Math., 53(1):71-77, 1993. MR 93j:76039
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Additional Information:
John
Etnyre
Affiliation:
Department of Mathematics, Stanford University, Stanford, California, 94305
Email:
etnyre@math.stanford.edu
Robert
Ghrist
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email:
ghrist@math.gatech.edu
DOI:
10.1090/S0002-9947-00-02651-9
PII:
S 0002-9947(00)02651-9
Keywords:
Tight contact structures,
Reeb flows,
Euler equations,
knots,
templates
Received by editor(s):
June 29, 1999
Posted:
August 8, 2000
Additional Notes:
JE supported in part by NSF Grant # DMS-9705949.
RG supported in part by NSF Grant # DMS-9971629.
Copyright of article:
Copyright
2000,
American Mathematical Society
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