zbMATH — the first resource for mathematics

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
Full Text:
References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.