Algebraic & Geometric Topology Volume 13, issue 5 (2013)

Download this article
	For screen
	For printing

Recent 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

DOI: 10.2140/agt.2013.13.2897

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