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Brownian motion can see a knot

Published online by Cambridge University Press:  24 October 2008

N. Th. Varopoulos
N. Th. Varopoulos
Affiliation:
Université de Paris VI
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Extract

Let be the 3-dimensional sphere and let be a smooth knot, i.e. K is the image of a regular diffeomorphism . We say that K is unknotted if there exists a diffeomorphism of onto the unit sphere of , that brings K to the equatorial circle otherwise we say that K is knotted.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 2 , March 1985 , pp. 299 - 309
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Carne, T. K.. Private communication.Google Scholar
[2]Deny, J.. Méthodes hilbertiennes en théorie du potentiel. In Potential Theory (C.I.M.E. 1969) (Edizioni Cremonese, 1970), 121–201.Google Scholar
[3]Gromov, M.. Groups of polynomial growth and expanding maps. Publ. Math. I.H.E.S. 53 (1981), 53–73.CrossRefGoogle Scholar
[4]Guivarc'h, Y., Keane, M. S. and Roynette, B.. Marches aléatoires sur les groupes de Lie. Lecture Notes in Mathematics, vol. 624 (Springer-Verlag, 1977).CrossRefGoogle Scholar
[5]Itô, K. and McKean, H. P.. Diffusion Processes and their Sample Paths. Die Grundlehren der math. Wissenschaften Band 125 (Springer-Verlag, 1965).Google Scholar
[6]Milnor, J.. A note on curvature and fundamental group. J. Differential Geom. 2 (1968), 1–7.CrossRefGoogle Scholar
[7]Neuwirth, L. P.. Knot Groups. Ann. of Math. Studies, no. 56 (Princeton University Press, 1965).Google Scholar
[8]Stillwell, J.. Classical Topology and Combinatorial Group Theory. Graduate Texts in Math., vol. 72 (Springer-Verlag, 1980).CrossRefGoogle Scholar
[9]Varopoulos, N. Th.. Brownian motion and transient groups. Ann. Inst. Fourier 33 (1983), 241–261.CrossRefGoogle Scholar
[10]Varopoulos, N. Th.. Brownian motion and random walks on manifolds. Ann. Inst. Fourier 34 (1984), 243–269.CrossRefGoogle Scholar
[11]Varopoulos, N. Th.. Ark. Mat. (to appear).Google Scholar
[12]Varopoulos, N. Th.. Groupes transients en analyse complexe. C.R. Acad. Sci. Paris 297 (1983), Sér. I, 585–586.Google Scholar
[13]Varopoulos, N. Th.. Chaînes de Markov et inégalités isopérimétriques. C.R. Acad. Sci. Paris 298 (1984), Sér. I, 233–236.Google Scholar
[14]Varopoulos, N. Th.. Potential theory and diffusion on Riemannian manifolds. In Conference on Harmonic Analysis in Honor of Antoni Zygmund, vol. II (Wadsworth, 1983), 821–837.Google Scholar
[15]Williams, D.. Diffusions, Markov Processes and Martingales (J. Wiley, 1979).Google Scholar