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The Jones-Witten invariant for flows on a 3-dimensional manifold. (English) Zbl 0808.57011
The authors generalize the Jones-Witten polynomial invariants for links on closed compact orientable 3-manifolds, for a vector field with an invariant measure. In the case of measures supported on a finite number of closed orbits and uniformly distributed along them, the new invariant is essentially Witten’s formula for links (the closed orbits) on the 3- manifold. Furthermore in the case that the gauge group (used in the definition) is abelian, explicit computations of these invariants are given by the authors.

MSC:
57N10 Topology of general \(3\)-manifolds (MSC2010)
37C10 Dynamics induced by flows and semiflows
57M25 Knots and links in the \(3\)-sphere (MSC2010)
37A99 Ergodic theory
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[1] [A] Arnold, V.I.: The asymptotic Hopf invariant and its applications. Materialy Vsesoyuznoi Shkoly po Differentsialnym Uraveneniyam s Beskonechnym Chislom Nezavisimyh i po Dynamicheskim Sisteman c Beskonechnym... (in Russian), 1973–74; English transl. in Sel. Math. Sov.5(4), 327–345 (1986)
[2] [At] Atiyah, M.: The geometry and physics of knots. Cambridge: Cambridge University Press 1990 · Zbl 0729.57002
[3] [AB] Atiyah, M., Bott, R.: The Yang Mills equations over Riemann surfaces. Phil. Trans. R. Soc. Lond. A308, 523–615 (1982) · Zbl 0509.14014
[4] [CFS] Cornfeld, I.P., Fomin, S.V., Sinai, Y.G.: Ergodic theory, Berlin, Heidelberg, New York: Springer 1982 · Zbl 0493.28007
[5] [D] Dolgacev, I.V.: Conic quotient singularities of complex surfaces. Funct. Anal. Appl.8(2), 160–161 (1974) · Zbl 0295.14017 · doi:10.1007/BF01078607
[6] [KC] Khesin, B.A., Chekanov, Yu.V.: Invariants of the Euler equations for ideal or barotropic hydrodynamics and superconductivity inD dimensions. Physica D40, 119–131 (1989) · Zbl 0820.58019 · doi:10.1016/0167-2789(89)90030-4
[7] [KN] Kobayashi, S., Nomizu, K.: Foundations of differential geometry, Vol. 1. New York: Wiley 1963 · Zbl 0119.37502
[8] [M] Milnor, J.: On the 3-dimensional Brieskorn manifoldsM(p,q,r). In: Knots, groups and 3-manifolds (Neuwirth, ed.) Ann. Math. Studies. Princeton, NJ: Princeton Univ. Press 1975, pp. 175–225
[9] [N] Nelson, E.: Topics in dynamics. I. Flows. Princeton: Princeton Univ. Press 1969 · Zbl 0197.10702
[10] [No] Novikov, S.P.: The analytic generalized Hopf invariant. Many-valued functionals (in Russian). Uspekhi Mat. Nauk (1984); English transl. in Russ. Math. Surv.39(5), 113–124 (1984)
[11] [O] Oseledec, V.I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc.19, 197–231 (1968)
[12] [P] Polyakov, A.M.: Fermi-Bose transmutations induced by gauge fields. Mod. Phys. Lett. A3, 325 (1988) · doi:10.1142/S0217732388000398
[13] [RSW] Ramadas, T.R., Singer, I.M., Weitsman, J.: Some comments on Chern-Simons gauge theory. Commun. Math. Phys.126, 409–420 (1989) · Zbl 0686.53066 · doi:10.1007/BF02125132
[14] [Sc] Schwartzman, S.: Asymptotic cycles. Ann. Math.66(2), 270–284 (1957) · Zbl 0207.22603 · doi:10.2307/1969999
[15] [Su] Sullivan, D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math.36, 225–255 (1976) · Zbl 0335.57015 · doi:10.1007/BF01390011
[16] [T] Tabachnikov, S.L.: Two remarks on the asymptotic Hopf invariant (in Russian). Funktsional’nyi Analiz i Ego Prilozheniya (1990); English transl. in Funct. Anal. and its Appl.24(1) (1990)
[17] [W] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351–399 (1989) · Zbl 0667.57005 · doi:10.1007/BF01217730
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