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On volume-preserving vector fields and finite-type invariants of knots

Published online by Cambridge University Press:  06 November 2014

R. KOMENDARCZYK  and
I. VOLIĆ
R. KOMENDARCZYK
Affiliation:
Tulane University, New Orleans, LA 70118, USA email rako@tulane.edu
I. VOLIĆ
Affiliation:
Wellesley College, Wellesley, MA 02481-8203, USA email ivolic@wellesley.edu
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Abstract

We consider the general non-vanishing, divergence-free vector fields defined on a domain in $3$-space and tangent to its boundary. Based on the theory of finite-type invariants, we define a family of invariants for such fields, in the style of Arnold’s asymptotic linking number. Our approach is based on the configuration space integrals due to Bott and Taubes.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 3 , May 2016 , pp. 832 - 859
Copyright
© Cambridge University Press, 2014 

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References

Akhmetiev, P. M.. On a new integral formula for an invariant of 3-component oriented links. J. Geom. Phys. 53(2) (2005), 180–196.CrossRefGoogle Scholar
Akhmetiev, P. M.. Quadratic helicities and the energy of magnetic fields. Proc. Steklov Inst. Math. 278(1) (2012), 10–21.CrossRefGoogle Scholar
Altschüler, D. and Freidel, L.. On universal Vassiliev invariants. Comm. Math. Phys. 170(1) (1995), 41–62.Google Scholar
Arnold, V. I.. The asymptotic Hopf invariant and its applications. Selecta Math. Sov. 5(4) (1986), 327–345, Selected translations.Google Scholar
Arnold, V. I.. Arnold’s Problems. Springer, Berlin, 2004. Translated and revised edition of the 2000 Russian original, with a preface by V. Philippov, A. Yakivchik, and M. Peters.Google Scholar
Arnold, V. I. and Khesin, B. A.. Topological Methods in Hydrodynamics (Applied Mathematical Sciences, 125). Springer, New York, 1998.Google Scholar
Baader, S.. Asymptotic Rasmussen invariant. C. R. Math. Acad. Sci. Paris 345(4) (2007), 225–228.CrossRefGoogle Scholar
Baader, S.. Asymptotic concordance invariants for ergodic vector fields. Comment. Math. Helv. 86(1) (2011), 1–12.Google Scholar
Baader, S. and Marché, J.. Asymptotic Vassiliev invariants for vector fields. Bull. Soc. Math. France 140(4) (2012), 569–582.Google Scholar
Bar-Natan, D.. On the Vassiliev knot invariants. Topology 34(2) (1995), 423–472.Google Scholar
Becker, M. E.. Multiparameter groups of measure-preserving transformations: a simple proof of Wiener’s ergodic theorem. Ann. Probab. 9(3) (1981), 504–509.Google Scholar
Billingsley, P.. Convergence of Probability Measures (Wiley Series in Probability and Statistics: Probability and Statistics), 2nd edn. John Wiley, New York, 1999. A Wiley–Interscience Publication.CrossRefGoogle Scholar
Bodecker, H. v. and Hornig, G.. Link invariants of electromagnetic fields. Phys. Rev. Lett. 92(3) (2004), 030406.Google Scholar
Bott, R. and Taubes, C.. On the self-linking of knots. J. Math. Phys. 35(10) (1994), 5247–5287.Google Scholar
Călugăreanu, G.. L’intégrale de Gauss et l’analyse des nœuds tridimensionnels. Rev. Math. Pures Appl. 4 (1959), 5–20.Google Scholar
Candel, A. and Conlon, L.. Foliations. I (Graduate Studies in Mathematics, 23). American Mathematical Society, Providence, RI, 2000.Google Scholar
Cantarella, J. and Parsley, J.. A new cohomological formula for helicity in ℝ2k+1 reveals the effect of a diffeomorphism on helicity. J. Geom. Phys. 60(9) (2010), 1127–1155.Google Scholar
Chmutov, S., Duzhin, S. and Mostovoy, J.. Introduction to Vassiliev Knot Invariants. Cambridge University Press, Cambridge, 2012.CrossRefGoogle Scholar
Contreras, G. and Iturriaga, R.. Average linking numbers. Ergod. Th. & Dynam. Sys. 19(6) (1999), 1425–1435.Google Scholar
Evans, N. W. and Berger, M. A.. A hierarchy of linking integrals. Topological Aspects of the Dynamics of Fluids and Plasmas (Santa Barbara, CA, 1991) (NATO Advanced Science Institutes Series E: Applied Sciences, 218). Kluwer Academic, Dordrecht, 1992, pp. 237–248.Google Scholar
Fadell, E. R. and Husseini, S. Y.. Geometry and Topology of Configuration Spaces (Springer Monographs in Mathematics). Springer, Berlin, 2001.CrossRefGoogle Scholar
Freedman, M. H.. Zeldovich’s neutron star and the prediction of magnetic froth. The Arnoldfest (Toronto, ON, 1997) (Fields Institute Communications, 24). American Mathematical Society, Providence, RI, 1999, pp. 165–172.Google Scholar
Freedman, M. H. and He, Z.-X.. Divergence-free fields: energy and asymptotic crossing number. Ann. of Math. (2) 134(1) (1991), 189–229.Google Scholar
Fulton, W. and MacPherson, R.. A compactification of configuration spaces. Ann. of Math. (2) 139(1) (1994), 183–225.CrossRefGoogle Scholar
Gambaudo, J. and Ghys, É.. Enlacements asymptotiques. Topology 36(6) (1997), 1355–1379.CrossRefGoogle Scholar
Gambaudo, J. M. and Ghys, É.. Signature asymptotique d’un champ de vecteurs en dimension 3. Duke Math. J. 106(1) (2001), 41–79.Google Scholar
Khesin, B. A.. Geometry of higher helicities. Mosc. Math. J. 3(3) (2003), 989–1011, 1200 (Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday).Google Scholar
Komendarczyk, R.. The third order helicity of magnetic fields via link maps. Comm. Math. Phys. 292(2) (2009), 431–456.Google Scholar
Komendarczyk, R.. The third order helicity of magnetic fields via link maps. II. J. Math. Phys. 51(12) (2010), 122702.Google Scholar
Kontsevich, M.. Vassiliev’s knot invariants. I. M. Gelfand Seminar (Advances in Soviet Mathematics, 16). American Mathematical Society, Providence, RI, 1993, pp. 137–150.Google Scholar
Kotschick, D. and Vogel, T.. Linking numbers of measured foliations. Ergod. Th. & Dynam. Sys. 23(2) (2003), 541–558.CrossRefGoogle Scholar
Laurence, P. and Stredulinsky, E.. Asymptotic Massey products, induced currents and Borromean torus links. J. Math. Phys. 41(5) (2000), 3170–3191.Google Scholar
Lieb, E. H. and Loss, M.. Analysis, 2nd edn (Graduate Studies in Mathematics, 14). American Mathematical Society, Providence, RI, 2001.Google Scholar
Melrose, R. B.. Differential analysis on manifolds with corners. Online notes available at http://www-math.mit.edu/∼rbm/book.html.Google Scholar
Moffatt, H. K.. The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35(1) (1969), 117–129.Google Scholar
Rivière, T.. High-dimensional helicities and rigidity of linked foliations. Asian J. Math. 6(3) (2002), 505–533.Google Scholar
Ruelle, D. and Sullivan, D.. Currents, flows and diffeomorphisms. Topology 14(4) (1975), 319–327.CrossRefGoogle Scholar
Sinha, D. P.. Manifold-theoretic compactifications of configuration spaces. Selecta Math. (N.S.) 10(3) (2004), 391–428.Google Scholar
Tempelman, A. A.. Ergodic theorems for general dynamical systems. Dokl. Akad. Nauk SSSR 176 (1967), 790–793.Google Scholar
Thurston, D. P.. Integral expressions for the Vassiliev knot invariants. Senior Thesis, Harvard University, April 1995, arXiv:math/9901110, 1999.Google Scholar
Vassiliev, V. A.. Complements of Discriminants of Smooth Maps: Topology and Applications (Translations of Mathematical Monographs, 98). American Mathematical Society, Providence, RI, 1992. Translated from the Russian by B. Goldfarb.Google Scholar
Verjovsky, A. and Vila Freyer, R. F.. The Jones–Witten invariant for flows on a 3-dimensional manifold. Comm. Math. Phys. 163(1) (1994), 73–88.CrossRefGoogle Scholar
Vogel, T.. On the asymptotic linking number. Proc. Amer. Math. Soc. 131(7) (2003), 2289–2297, (electronic).Google Scholar
Volić, I.. A survey of Bott–Taubes integration. J. Knot Theory Ramifications 16(1) (2007), 1–42.Google Scholar
Woltjer, L.. On hydromagnetic equilibrium. Proc. Natl Acad. Sci. USA 44 (1958), 833–841.Google Scholar