Multiplace generalizations of the Seifert form of a classical knot
Sbornik. Mathematics, Tome 44 (1983) no. 3, pp. 335-361 Cet article a éte moissonné depuis la source Math-Net.Ru

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On the fundamental group $\pi$ of a Seifert surface$A$ of a knot in the three-dimensional sphere, the author constructs, using the same scheme as for the Seifert form, a form $\pi^n\to\mathbf Z$, for $n=3,4,\dots$ . The role of linking coefficient is played here by suitably chosen integral representatives of Milnor residues. It is shown that the form $\pi^3\to\mathbf Z$ can obstruct invertibility, ribbonness and two-sided null-cobordancy of the knot $\partial A$ (even when there is no obstruction by the Seifert form itself). Figures: 5. Bibliography: 17 titles.
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     title = {Multiplace generalizations of the {Seifert} form of a~classical knot},
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V. G. Turaev. Multiplace generalizations of the Seifert form of a classical knot. Sbornik. Mathematics, Tome 44 (1983) no. 3, pp. 335-361. http://geodesic.mathdoc.fr/item/SM_1983_44_3_a6/

[1] Andreadakis S., “On the automorphisms of free groups and free nilpotent groups”, Proc. London Math. Soc., (3), 15:2 (1965), 239–268 | DOI | MR | Zbl

[2] Baumslag G., “Lecture notes on nilpotent groups”, Reg. Conf. Ser. Math., 1971, no. 2 | MR

[3] Casson A. J., Gordon C. McA., Cobordism of classical knots, Preprint, 1975 | MR

[4] Erie D., “Die quadratische Form eines Knotens und ein Satz über Knotenmannigfaltigkeiten”, J. reine und angew. Math., 236 (1969), 174–218 | MR

[5] Fox R. H., Chen K. T., Lyndon R. C., “Free differential calculus IV. The quotient groups of the lower central series”, Ann. Math., 68 (1958), 81–95 | DOI | MR

[6] Gruenberg K. W., Cohomological topics in group theory, Lecture Notes in Math., 143, 1970 | MR | Zbl

[7] Levine J., “Knot cobordism groups in codimension two”, Comment. Math. Helv., 44 (1969), 229–244 | DOI | MR | Zbl

[8] Milnor J. W., Isotopy of links. Algebr. Geometry and Topology, Univ. Press, Princeton, N. J., 1957 | MR

[9] Milnor J. W., “Infinite cyclic coverings”, Conf. on the Topology of Manifolds (Michigan, 1967), Mass., Boston, 1968, 115–133 | MR

[10] Quillen D. G., “On the associated graded ring of a group ring”, J. Algebra, 10:4 (1968), 411–418 | DOI | MR | Zbl

[11] Seifert H., “Über das Geschlecht von Knoten”, Math. Ann., 110 (1934), 571–592 | DOI | MR

[12] Stammbach U., Homology in Group Theory, Lecture Notes in Math., 359, 1973 | MR | Zbl

[13] Sumners D. W., “Invertible knot cobordisms”, Comm. Math. Helv., 46:2 (1971), 240–256 | DOI | MR | Zbl

[14] Trotter H. F., “Noninvertible knots exist”, Topology, 2 (1963), 341–358 | DOI | MR

[15] Trotter H. F., “On $S$-equivalence of Seifert matrices”, Invent., Math., 20:3 (1973), 173–207 | DOI | MR | Zbl

[16] Turaev V. G., “Invarianty Milnora i proizvedeniya Massi”, Zap. nauchn. sem. LOMI, 66 (1976), 189–203 | MR | Zbl

[17] Turaev V. G., “Peresecheniya petel v dvumernykh mnogoobraziyakh”, Matem. sb., 106(148) (1972), 566–588 | MR