A survey on the differential and symplectic geometry of linking numbers

Milan Journal of Mathematics

Volumn 74, Issue 1, 2006, Pages 139-197

  Spera, Mauro ^a  

^a UNIVERSITY OF PADOVA   (Italy)

Author keywords

Artin's braid groups; Chen iterated integrals; Chern Simons action; Geometric and topological methods of hydrodynamics; Geometric quantization; Magnetic relaxation; Maslov theory; Ordinary and higher order linking numbers

Indexed keywords

EID: 33751540992     PISSN: 14249286     EISSN: 14249294     Source Type: Journal    
DOI: 10.1007/s00032-006-0061-5     Document Type: Article

References (118)

1. A. Adem, D. Cohen and F.R. Cohen, On representations and K-theory of the braid groups. Math. Ann. 326 (2003), 515-542.
2. P. Akhmetiev, On a new integral formula for an invariant of 3-component oriented links. J. Geom. Phys. 53 (2005), 180-196.
3. P. Akhmetiev and A. Ruzmaikin, A fourth-order topological invariant of magnetic or vortex lines. J. Geom. Phys. 15 (1995), 95-101.
4. S. Albeverio and A. Sen-Gupta, A Mathematical Construction of the Non-Abelian Chern-Simons Functional Integral. Commun. Math. Phys. 186 (1997), 563-579.
5. P.K. Aravind, Borromean entanglement of the GHZ state in "Potentiality, Entanglement and Passion-at-a-Distance". R.S. Cohen, M. Home and J. Stachel eds. Kluwer Academic Publishers, Boston, 1997.
6. V.I. Arnol'd, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16(1966), 316-361.
7. V.I. Arnol'd, The asymptotic Hopf invariant and its applications. Proc. Summer School in Diff.Equations at Dilizhan, 1973 (1974) Erevan (in Russian); English transl. in Sci. Math. Sov. 5 (1986), 327-345.
8. V.I. Arnol'd and B. Khesin, Topological Methods in Hydrodynamics. Springer, Berlin, 1998.
9. E. Artin, Theorie der Zöpfe. Abh. Math. Sem. Hamburg Univ. 4 (1925), 42-72;
10. Theory of braids Ann. Math. 48 (1947), 101-126.
11. A. Ashtekar and A. Corichi, The Gauss linking number and the electromagnetic uncertainty principle. Phys. Rev. D (3) 56 (1997), 2073-2079.
12. R. Gambini and J. Pullin, The Gauss linking number in quantum gravity. Knots and Quantum Gravity, Riverside, CA, 1993, 63-75, Oxford Lecture Ser. Math. Appl. 1, Oxford Univ. Press, New York, 1994.
13. M. Atiyah, The Geometry and Physics of Knots. Lezioni Lincee, Cambridge University Press, Cambridge, 1990.
14. D. Bar-Natan, On the Vassiliev knot invariants. Topology 34 (1995), 423-472.
15. D. Bar-Natan, Vassiliev homotopy string link invariants. J. Knot Theory and Its Ramifications 4 (1995), 13-32.
16. A. Benvegnù and M. Spera, On Uncertainty, Braiding and Entanglement in Geometric Quantum Mechanics. Preprint (R.T.101-DMMMSA, Padova), 2006.
17. M. Berger, Third order link invariants. J. Phys. A: Math.Gen. 23 (1990), 2787-2793.
18. M. Berger, Third order braid invariants. J. Phys. A: Math.Gen. 24 (1991), 4027-4036.
19. M. Berger, Energy-Crossing Number relations for Braided Magnetic Fields. Phys. Rev. Lett. 70 (1992), 705-708.
20. M. Berger, Hamiltonian dynamics generated by Vassiliev invariants. J. Phys. A: Math.Gen. 34 (2001), 1363-1374.
21. M. Berger, Topological Invariants in braid theory. Lett. Math. Phys. 55 (2001), 181-192.
22. A. Besana, Framed knots, Lagrangian submanifolds and geometric quantization. Ph.D. Thesis, Università degli Studi di Milano, 2004.
23. A. Besana and M. Spera, On some symplectic aspects of knot framings. J. Knot Theory Ram. (to appear).
24. J. Birman, Braids, Links and Mapping Class Groups. Ann. Math. Stud. 82, Princeton, NJ, 1974.
25. C.P. Black, A Mathematical Verification of the Existence of Strings of Circulation in Superfluid Films on Porous Media. Preprint.
26. H. Bodecker and G. Hornig, Link invariants of electromagnetic fields. Phys. Rev. Lett. 92 030406, (2004).
27. R. Bott and L.T. Tu, Differential Forms in Algebraic Topology. Springer, Berlin, Heidelberg, New York, 1982.
28. J.L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization. Birkhäuser, Basel, 1993.
29. R.V. Chacon and A.F. Fomenko, Stokes' Formula for Lie Algebra Valued Connection and Curvature Forms. Adv. Math. 88 (1991), 258-300.
30. T. Chakraborty and P.Pietiläinen, The Quantum Hall Effects, Fractional and Integral. Springer, Berlin, 1995.
31. F. Wilczek, Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49 (1982), 957-959.
32. R.B. Laughlin, Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionallly Charged Excitations. Phys. Rev. Lett. 50 (1983), 1395-1402.
33. B.I. Halperin, Quasiparticles and the Hierarchy of Fractional Quantized Quantum Hall Effect. Phys. Rev. Lett. 52 (1983), 1583-1586; Erratum (1984), 2390.
34. K.-T. Chen, Iterated integrals of differential forms and loop space homology. Ann. Math. 39 (1973), 217-246.
35. K.-T. Chen, Isotopy invariants of links. Ann. Math. 56 (1952), 343-353.
36. K.-T. Chen, Iterated path integrals. Bull. Am. Math. Soc. 83 (1977), 831-879.
37. K.-T. Chen, Collected Papers of K.-T. Chen. P.Tondeur and R.Hain eds. Contemporary Mathematicians, Birkäuser, Boston, MA 2001.
38. D. Chruściński and A. Jamiołkowski, Geometric Phases in Classical and Quantum Mechanics. Birkhäuser, Boston, 2004.
39. R. Coquereaux and K. Pilch, String Structures on Loop Bundles. Commun.Math. Phys. 120 (1989), 353-378.
40. G. de Rham, Variétés différentiables. Hermann, Paris, 1954.
41. J.D. Bollard and C.N. Friedman, Product Integration. E.M.A. Addison-Wesley P.C. London, Amsterdam, 1979.
42. D.G. Ebin and J.E. Marsden, Groups of diffeomorphisms and the motion of incompressible fluids. Ann. Math. 92 (1970), 102-163.
43. M. Epple, Topology, Matter, and Space, I: Topological Notions in the 19th-century Natural Philosophy. Arch. Hist. Exact Sci. 52 (1998), 297-392.
44. N.W. Evans and M.A. Berger, A hierarchy of linking integrals, in Topological Aspects of the Dynamics of Fluids and Plasmas. H.K. Moffatt et al. eds. Kluwer, Dordrecht, The Netherlands, 1992, 237-248.
45. R.A. Fenn, Techniques of Geometric Topology. London Mathematical Society, Lecture Notes Series 57 Cambridge University Press, Cambridge, 1983.
46. R. Feynman, in Progress in Low Temperature Physics, Vol.1, edited by C.J. Gorter, Amsterdam, 1955.
47. M. Freedman, A note on topology and magnetic energy in incompressible perfectly conducting fluids. 194 (1988), 549-554.
48. M. Freedman and Z.-X. He, Divergence-free fields: energy and asymptotic crossing number. Ann. Math. 134 (1991), 189-229.
49. C.F. Gauß, Werke. Königichlen Gesellshaft der Wissenshaften zu Göttingen, Leipzig-Berlin (1867), vol. 5.
50. G. Goldin, Parastatistics, θ-statistics, and Topological Quantum Mechanics from Unitary Representations of Diffeomorphism Groups. Proceedings of the XV International Conference on Differential Geometric Methods in Physics, H.D Doebner and J.D.Henning eds. World Scientific, Singapore, 1987, 197-207.
51. G. Goldin, R. Menikoff and D. Sharp, Diffeomorphism Groups and Quantized Vortex Filaments. Phys. Rev. Lett. 58 (1987), 2162-2164.
52. G. Goldin, R. Menikoff and D. Sharp, Quantum Vortex Configurations in Three Dimensions. Phys. Rev. Lett. 67 (1991), 3499-3502.
53. P. Griffiths and J. Harris, Principles of Algebraic Geometry. J.Wiley & Sons, New York, 1978.
54. E. Guadagnini, The Link Invariants of the Chern-Simons Field Theory. W.de Gruyter, Berlin, 1993.
55. T. Guénault, Basic Superfluids. Taylor & Francis, London, 2003.
56. V. Guillemin and S. Sternberg, Geometric Asymptotics. AMS, Providence, RI, 1977.
57. R. Hain, Iterated Integrals, Intersection Theory and Link Groups. Topology 24 (1985), 45-66; Erratum: Topology 25 (1986), 585-586.
58. R. Hain, The Geometry of the Mixed Hodge Structure on the Fundamental Group. Proc. Symp. Pure Math. 46 (1987), 247-282.
59. L. Hörmander, Fourier Integral operators I. Acta Math. 127 (1971), 79.
60. G. Hornig and C. Mayer, Towards a third-order topological invariant for magnetic fields. J. Phys. A (Math. Gen.) 35 (2002), 3945-3959.
61. L.H. Kauffman, Knots and Physics. 3rd edition World Scientific, Singapore, 2001.
62. L.H. Kauffman, Witten's integral and the Kontsevich integral, in Particles, fields, and gravitation. Lodz, 1998, 368-381,
63. AIP Conf. Proc. 453, Woodbury, NY, 1998.
64. L.H. Kauffman, Functional integration and the theory of knots. J. Math. Phys. 36 (1995), 2402-2429.
65. B. Khesin, Topological Fluid Dynamics. Notices of the Am. Math. Soc. 52 (2005), 9-19.
66. A. Kirillov, Elements of the Theory of Representations. Springer, Berlin, 1976.
67. A.Kirillov, Lectures on the orbit method. AMS, Providence, RI, 2004.
68. T. Kohno, Conformal Field Theory and Topology. AMS, Providence, RI, 2002.
69. M. Kontsevich, Vassiliev's knot invariants. Adv. Sov. Math. 16, Part 2 (1993), 137-150, AMS Providence, RI.
70. B. Kostant, Quantization and unitary representations, in Lectures in modern analysis and applications III, Lecture Notes in Mathematics 170 (1970), Springer, Heidelberg, 87-208.
71. A. Kriegl and P.W. Michor, The Convenient Setting of Global Analysis. AMS, Providence, RI, 1997.
72. E.A. Kuznetsov and A.V. Mikhailov, On the topological meaning of canonical Clebsch variables. Phys. Lett. A 77 (1980), 37-38.
73. H. Lamb, Hydrodynamics. Dover, New York, 1945.
74. W. Magnus, A. Karras and D. Solitar, Combinatorial Group Theory. Dover, New York, 1976
75. V.P. Maslov, Théorie des Perturbations et Méthodes Asymptotiques. Ed. de l'Université de Moscou, 1965 (en russe). Traduction française: Dunod - Gauthier-Villars, Paris, 1971.
76. W. Massey, Higher order linking numbers. Proc. Conf. on Algebraic Topology, 1969, 174-205.
77. Reprinted in Journal of Knot Theory and Its Ramifications 7 (1998), 393-414.
78. D. McDuff and D. Salamon, Introduction to Symplectic Topology. Clarendon Press, Oxford, 1998.
79. J. Marsden and A. Weinstein, Coadjoint orbits, vortices, Clebsch variables for incompressible fluids. Physica 7 D (1983), 305-323.
80. J. Milnor, Link Groups. Ann. Math. 59 (1954), 177-195.
81. J. Milnor, Isotopy of links. Proc. Symposium in honor of S. Lefschetz, 280-306, Princeton University Press, Princeton, New York, 1957.
82. 4He Film Adsorbed on Porous Materials. Progr. Theor. Phys. 80 (1988), 397-416.
83. H.K. Moffatt, The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35 (1969), 116-144.
84. H.K. Moffatt, The energy spectrum of knots and links. Nature 347 (1990), 367-369.
85. H.K. Moffatt and R.L. Ricca, Helicity and the Cǎlugǎreanu invariant. Proc. R. Soc. Lond. A 439 (1992), 411-429.
86. M.I. Monastyrsky and V.S. Retakh, Topology of Linked Defects in Condensed Matter. Commun. Math. Phys. 103 (1986), 445-459.
87. G.D. Mostow, Braids, hypergeometric functions, and lattices. Bull. Am. Math. Soc. 16 (1987), 225-246.
88. C. Nash, Differential Topology and Quantum Field Theory. Academic Press, London, 1991.
89. L. Onsager, Suppl. Nuovo Cimento 6 (1949), 279.
90. P. Papi and C. Procesi, Invarianti di nodi. Quaderno U.M.I. 45, Pitagora, Bologna, 1998 (in Italian).
91. V. Penna, M. Rasetti and M. Spera, Iterated path integrals realization of quantum vortex currents: construction of the topological invariants. Intl. J. Mod. Phys. B 4 (1990), 1289-1315.
92. V. Penna, M. Rasetti and M. Spera, Quantum Dynamics of 3-D Vortices, in "Secondary Calculus and Cohomological Physics". M. Henneaux, J. Krasil'shchic and A. Vinogradov eds. Contemporary Mathematics 219, AMS, 1998, 173-193.
93. V. Penna and M. Spera, A geometric approach to quantum vortices. J. Math. Phys. 30 (1989), 2778-2784.
94. V. Penna and M. Spera, On coadjoint orbits of rotational perfect fluids. J. Math. Phys. 33 (1992), 901-909.
95. V. Penna and M. Spera, Remarks on quantum vortex theory on Riemann surfaces. J. Geom. Phys. 27 (1998), 99-112.
96. V. Penna and M. Spera, String limit of vortex current algebra. Phys. Rev. B 62 (2000), 14547-14553.
97. V. Penna and M. Spera, Higher order linking numbers, curvature and holonomy. J. Knot Theory Ram. 11 (2002), 701-723.
98. R. Porter, Milnor's μ̄-Invariants and Massey Products. Trans. Am. Math. Soc. 257 (1980), 39-71.
99. M. Rasetti and T. Regge, Physica 80 A (1975), 217.
100. R.L. Ricca, Applications of knot theory in fluid mechanics. Banach Center Publications 42, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1988, 321-346.
101. R.L. Ricca, Geometric and topological aspects of vortex motion, in "An Introduction to the Geometry and Topology of Fluid Flows". R.L. Ricca ed. Kluwer Academic Publishers, The Netherlands, 2001, 203-228.
102. R.L. Ricca and M.A. Berger, Topological ideas and fluid mechanics. Physics Today, AIP, 1996, 28-34.
103. D. Rolfsen, Knots and Links. Publish or Perish, Berkeley, 1976
104. L. Schlesinger, Parallel Verschiebung und Krümmungstensor. Math. Ann.99 (1928), 413-434.
105. J-M. Souriau, Structure des Systèmes Dynamiques. Dunod, Paris, 1970.
106. J.N. Tavares, Chen Integrals, Generalized Loops and Loop Calculus. Intl. J. Mod. Phys.A 9 (1994), 4511-4548.
107. M.E. Taylor, Partial Differential Equations III: Nonlinear Equations. Springer, New York, 1996.
108. V.G. Turaev, Milnor invariants and Massey products. Zap. Semin. Lomi 66 (1976), 189-203 (in Russian);
109. English translation: J. Soviet Math. 12 (1979), 128-137.
110. A. Tyurin, Quantization, Classical and Quantum Field Theory and Theta Functions. CRM Monograph Series, AMS, Providence, RI, 2003.
111. V. Vassiliev, Complements of Discriminants of smooth maps: Topology and Applications. AMS, Providence, 1992.
112. A. Verjovsky and R.F. Vila Freyer, The Jones-Witten Invariant for Flows on a 3-Dimensional Manifold. Commun. Math. Phys. 163 (1994), 73-88.
113. V. Vladimirov, Distributions en Physique Mathématique, French translation. Éd. MIR, Moscou, 1974.
114. A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds. Adv. Math. 6 (1971), 329-346.
115. E. Witten, Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121 (1989), 351-399.
116. N. Woodhouse, Geometric Quantization. 2nd Edition, Clarendon Press, Oxford, 1992.
117. Y. Wu, On the topological phases in a planar N-point vortex system. J. Math. Phys. 33 (1992), 1612-1617.
118. Y. Wu, On the quantization of the planar N-point vortex problem. J. Math. Phys. 34 (1993), 2342-2352.