Volume 13, issue 5 (2013)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
 
Other MSP Journals
Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows, part II

Dennis DeTurck, Herman Gluck, Rafal Komendarczyk, Paul Melvin, Haggai Nuchi, Clayton Shonkwiler and David Shea Vela-Vick

Algebraic & Geometric Topology 13 (2013) 2897–2923
Bibliography
1 P M Akhmetiev, On a new integral formula for an invariant of $3$–component oriented links, J. Geom. Phys. 53 (2005) 180 MR2110831
2 P Akhmetiev, A Ruzmaikin, A fourth-order topological invariant of magnetic or vortex lines, J. Geom. Phys. 15 (1995) 95 MR1310943
3 V I Arnold, B A Khesin, Topological methods in hydrodynamics, Applied Mathematical Sciences 125, Springer (1998) MR1612569
4 D Auckly, L Kapitanski, Analysis of $S^2$–valued maps and Faddeev's model, Comm. Math. Phys. 256 (2005) 611 MR2161273
5 M A Berger, Third-order link integrals, J. Phys. A 23 (1990) 2787 MR1062985
6 M A Berger, Third-order braid invariants, J. Physics A 24 (1991) 4027 MR1126646
7 H v. Bodecker, G Hornig, Link invariants of electromagnetic fields, Phys. Rev. Lett. 92 (2004) 030406 MR2068408
8 P Cromwell, E Beltrami, M Rampichini, The Borromean rings, Math. Intelligencer 20 (1998) 53 MR1601839
9 D DeTurck, H Gluck, R Komendarczyk, P Melvin, C Shonkwiler, D S Vela-Vick, Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links, Mat. Contemp. 34 (2008) 251 MR2588614
10 D DeTurck, H Gluck, R Komendarczyk, P Melvin, C Shonkwiler, D S Vela-Vick, Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows, J. Math. Phys. 54 (2013) 013515
11 N W Evans, M A Berger, A hierarchy of linking integrals, from: "Topological aspects of the dynamics of fluids and plasmas" (editors H K Moffatt, G M Zaslavsky, P Comte, M Tabor), NATO Adv. Sci. Inst. Ser. E Appl. Sci. 218, Kluwer Acad. Publ. (1992) 237 MR1232234
12 C F Gauss, Integral formula for linking number, from: "Werke Ergänzungsreihe, Band V" (editor C A F Peters), Georg Olms Verlag (1975) 605
13 E Guadagnini, M Martellini, M Mintchev, Wilson lines in Chern–Simons theory and link invariants, Nuclear Phys. B 330 (1990) 575 MR1043394
14 H Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, Math. Ann. 104 (1931) 637 MR1512691
15 G Hornig, C Mayer, Towards a third-order topological invariant for magnetic fields, J. Phys. A 35 (2002) 3945 MR1910658
16 B A Khesin, Geometry of higher helicities, Mosc. Math. J. 3 (2003) 989 MR2078570
17 R Komendarczyk, The third order helicity of magnetic fields via link maps, Comm. Math. Phys. 292 (2009) 431 MR2544738
18 R Komendarczyk, The third order helicity of magnetic fields via link maps, II, J. Math. Phys. 51 (2010) 122702 MR2779587
19 P Laurence, E Stredulinsky, Asymptotic Massey products, induced currents and Borromean torus links, J. Math. Phys. 41 (2000) 3170 MR1755498
20 L Leal, Link invariants from classical Chern–Simons theory, Phys. Rev. D 66 (2002) 125007 MR1959509
21 L Leal, J Pineda, The topological theory of the Milnor invariant $\overline\mu(1,2,3)$, Modern Phys. Lett. A 23 (2008) 205 MR2392325
22 W S Massey, Some higher order cohomology operations, from: "Symposium internacional de topología algebraica" (editor N C Flores), Universidad Nacional Autónoma de México (1958) 145 MR0098366
23 W S Massey, Higher order linking numbers, from: "Conf. on Algebraic Topology", Univ. of Illinois at Chicago Circle (1969) 174 MR0254832
24 J Milnor, Link groups, Ann. of Math. 59 (1954) 177 MR0071020
25 H K Moffatt, The degree of knottedness of tangled vortex lines, J. Fluid Mech. 35 (1969) 117
26 M I Monastyrsky, V S Retakh, Topology of linked defects in condensed matter, Comm. Math. Phys. 103 (1986) 445 MR832921
27 L Pontrjagin, A classification of mappings of the three-dimensional complex into the two-dimensional sphere, Rec. Math. [Mat. Sbornik] N. S. 9 (51) (1941) 331 MR0004780
28 T Rivière, High-dimensional helicities and rigidity of linked foliations, Asian J. Math. 6 (2002) 505 MR1946345
29 A Ruzmaikin, P Akhmetiev, Topological invariants of magnetic fields, and the effect of reconnections, Phys. Plasmas 1 (1994) 331 MR1269751
30 J H C Whitehead, An expression of Hopf's invariant as an integral, Proc. Nat. Acad. Sci. USA 33 (1947) 117 MR0020255
31 L Woltjer, A theorem on force-free magnetic fields, Proc. Nat. Acad. Sci. USA 44 (1958) 489 MR0096542