Generalized Gauss maps and integrals for three-component links: toward higher helicities for magnetic fields and fluid flows. II. (English) Zbl 1348.57007

Summary: We describe a new approach to triple linking invariants and integrals, aiming for a simpler, wider and more natural applicability to the search for higher order helicities.
To each three-component link in Euclidean 3-space, we associate a generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. This generalized Gauss map is a natural successor to Gauss’s original map from the 2-torus to the 2-sphere. Like its prototype, it is equivariant with respect to orientation-preserving isometries of the ambient space, attesting to its naturality and positioning it for application to physical situations.
When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number which is a natural successor to the classical Gauss integral for the pairwise linking numbers, with an integrand invariant under orientation-preserving isometries of the ambient space. This new integral is patterned after J H C Whitehead’s integral formula for the Hopf invariant, and hence interpretable as the ordinary helicity of a related vector field on the 3-torus.
For part I see [J. Math. Phys. 54, No. 1, 013515, 48 p. (2013; Zbl 1328.57007)].


57M25 Knots and links in the \(3\)-sphere (MSC2010)
76B99 Incompressible inviscid fluids
78A25 Electromagnetic theory (general)


Zbl 1328.57007
Full Text: DOI


[1] P M Akhmetiev, On a new integral formula for an invariant of \(3\)-component oriented links, J. Geom. Phys. 53 (2005) 180 · Zbl 1070.57007
[2] P Akhmetiev, A Ruzmaikin, A fourth-order topological invariant of magnetic or vortex lines, J. Geom. Phys. 15 (1995) 95 · Zbl 0836.57005
[3] V I Arnold, B A Khesin, Topological methods in hydrodynamics, Applied Mathematical Sciences 125, Springer (1998) · Zbl 0902.76001
[4] D Auckly, L Kapitanski, Analysis of \(S^2\)-valued maps and Faddeev’s model, Comm. Math. Phys. 256 (2005) 611 · Zbl 1082.58017
[5] M A Berger, Third-order link integrals, J. Phys. A 23 (1990) 2787 · Zbl 0711.57008
[6] M A Berger, Third-order braid invariants, J. Physics A 24 (1991) 4027 · Zbl 0747.57002
[7] H v. Bodecker, G Hornig, Link invariants of electromagnetic fields, Phys. Rev. Lett. 92 (2004) 030406 · Zbl 1267.78004
[8] P Cromwell, E Beltrami, M Rampichini, The Borromean rings, Math. Intelligencer 20 (1998) 53 · Zbl 0918.01017
[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 · Zbl 1194.57007
[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 · Zbl 1328.57007
[11] N W Evans, M A Berger, A hierarchy of linking integrals (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 · Zbl 0799.57004
[12] C F Gauss, Integral formula for linking numberanzungsreihe, 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 · Zbl 0957.81536
[14] H Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, Math. Ann. 104 (1931) 637 · Zbl 0001.40703
[15] G Hornig, C Mayer, Towards a third-order topological invariant for magnetic fields, J. Phys. A 35 (2002) 3945 · Zbl 1040.76067
[16] B A Khesin, Geometry of higher helicities, Mosc. Math. J. 3 (2003) 989 · Zbl 1156.55300
[17] R Komendarczyk, The third order helicity of magnetic fields via link maps, Comm. Math. Phys. 292 (2009) 431 · Zbl 1206.57007
[18] R Komendarczyk, The third order helicity of magnetic fields via link maps, II, J. Math. Phys. 51 (2010) 122702 · Zbl 1314.57007
[19] P Laurence, E Stredulinsky, Asymptotic Massey products, induced currents and Borromean torus links, J. Math. Phys. 41 (2000) 3170 · Zbl 0971.57019
[20] L Leal, Link invariants from classical Chern-Simons theory, Phys. Rev. D 66 (2002) 125007
[21] L Leal, J Pineda, The topological theory of the Milnor invariant \(\overline\mu(1,2,3)\), Modern Phys. Lett. A 23 (2008) 205 · Zbl 1141.81330
[22] W S Massey, Some higher order cohomology operations (editor N C Flores), Universidad Nacional Autónoma de México (1958) 145 · Zbl 0123.16103
[23] W S Massey, Higher order linking numbers, Univ. of Illinois at Chicago Circle (1969) 174 · Zbl 0212.55904
[24] J Milnor, Link groups, Ann. of Math. 59 (1954) 177 · Zbl 0055.16901
[25] H K Moffatt, The degree of knottedness of tangled vortex lines, J. Fluid Mech. 35 (1969) 117 · Zbl 0159.57903
[26] M I Monastyrsky, V S Retakh, Topology of linked defects in condensed matter, Comm. Math. Phys. 103 (1986) 445 · Zbl 0588.57009
[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 · JFM 67.0736.01
[28] T Rivière, High-dimensional helicities and rigidity of linked foliations, Asian J. Math. 6 (2002) 505 · Zbl 1045.57017
[29] A Ruzmaikin, P Akhmetiev, Topological invariants of magnetic fields, and the effect of reconnections, Phys. Plasmas 1 (1994) 331 · Zbl 0836.57005
[30] J H C Whitehead, An expression of Hopf’s invariant as an integral, Proc. Nat. Acad. Sci. USA 33 (1947) 117 · Zbl 0030.07902
[31] L Woltjer, A theorem on force-free magnetic fields, Proc. Nat. Acad. Sci. USA 44 (1958) 489 · Zbl 0081.21703
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