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Nombres de Milnor d’un entrelacs Brunnien. (Milnor numbers of Brunnian links). (French) Zbl 0581.57004
A link in the 3-sphere is said to be Brunnian if any of the sublinks which are obtained by removing one of its components is trivial. The authors introduce an invariant for p-component Brunnian links, which is a non-commutative polynomial. Using this polynomial, they compute the Milnor number \(\mu\) (1,...,p) and the Massey product associated to such a link. In particular, they recover a simple proof of the connection between Milnor numbers and Massey products, in the case of Brunnian links. In the course of the proofs, they also give a geometrical algorithm to determine their polynomial invariant, and to describe a Massey system for a Brunnian link.
Reviewer: F.Bonahon

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
55S30 Massey products
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[1] CHEN (K. T.) . - Algebras of iterated path integrals and fundamental groups . Trans. Amer. Math. Soc., vol. 156, 1971 , p. 359-379. MR 43 #1069 | Zbl 0217.47705 · Zbl 0217.47705 · doi:10.2307/1995617
[2] CHEN (K. T.) . - Isotopy invariants of links . Annals of Math. (2) vol. 56, 1952 , p. 343-353. MR 14,193g | Zbl 0049.40403 · Zbl 0049.40403 · doi:10.2307/1969803
[3] EDWARDS (S. F.) . - Statistical mechanics with topological constaints I et II . Proc. phys. soc. (Grande-Bretagne), 1967 , vol. 91 et série 2, 1968 , vol. 1. Zbl 0181.57002 · Zbl 0181.57002 · doi:10.1088/0305-4470/1/1/303
[4] FENN (R.) et TAYLOR (P.) . - Generalised Borromean ring and higher order linking numbers (à paraître) .
[5] FOX (R. H.) . - Free differential calculus I. Ann. of Math., t. 57, 1953 , p. 547-560. MR 14,843d | Zbl 0050.25602 · Zbl 0050.25602 · doi:10.2307/1969736
[6] GOLDSMITH (D.) . - A linking invariant of classical link concordance . Knot theory. Proceedings. Plans sur Bex, 1977 , (685). MR 80h:57005 | Zbl 0394.57006 · Zbl 0394.57006
[7] KRAINES (D.) . - Massey Higher products . Trans. Amer. Math. Soc., vol. 124, 1966 , p. 431-449. MR 34 #2010 | Zbl 0146.19201 · Zbl 0146.19201 · doi:10.2307/1994385
[8] LAUFER (H. B.) . - Some numerical link invariants . Topology, vol. 10, 1971 . MR 42 #8473 | Zbl 0192.60402 · Zbl 0192.60402 · doi:10.1016/0040-9383(71)90034-6
[9] MASSEY (W. S.) . - Some higher order cohomology operations , p. 145-154. Symposium International de Topologia Algebraica, Mexico City, 1958 . MR 20 #4826 | Zbl 0123.16103 · Zbl 0123.16103
[10] MASSEY (W. S.) . - Higher order linking numbers . Conf. on algebraic topology (Univ. of Illinois at Chicago circle, 1968 ). Univ. of Chicago at Chicago circle, Chicago, Illinois, p. 174-205. MR 40 #8039 | Zbl 0212.55904 · Zbl 0212.55904
[11] MAY (J. P.) . - Matric Massey products . J. Algebra, vol. 12, 1969 , p. 533-568. MR 39 #289 | Zbl 0192.34302 · Zbl 0192.34302 · doi:10.1016/0021-8693(69)90027-1
[12] MILNOR (J.) . - Link groups . Annals of Math., (2), 1954 , p. 177-195. MR 17,70e | Zbl 0055.16901 · Zbl 0055.16901 · doi:10.2307/1969685
[13] MILNOR (J.) . - Isotopy of links . Algebraic geometry and topology. A symposium in the honor of S. Lefschetz, p. 280-306. Princeton Univ. press. Princeton, New Jersey, 1957 . MR 19,1070c | Zbl 0080.16901 · Zbl 0080.16901
[14] MURASUGI (K.) . - On Milnor’s invariant for links , Trans. Amer. Math. Soc., vol. 124, 1966 , p. 94-110. MR 33 #6611 | Zbl 0178.57602 · Zbl 0178.57602 · doi:10.2307/1994437
[15] MURASUGI (K.) . - On Milnor’s invariant for links II . Trans. Amer. Math. Soc., vol. 148, 1970 , p. 41-61. MR 41 #4519 | Zbl 0197.20702 · Zbl 0197.20702 · doi:10.2307/1995036
[16] MURASUGI (K.) . - Nilpotent coverings of links and Milnor’s invariant . Preprint. · Zbl 0579.57004
[17] O’NEILL (E. J.) . - Higher order Massey Products and links . Trans. Amer. Math. Soc., vol. 248, 1979 , p. 37-66. MR 80h:55016 | Zbl 0407.55016 · Zbl 0407.55016 · doi:10.2307/1998736
[18] PORTER (R.) . - Milnor’s \mu invariants and Massey products . Trans. of the Amer. Math. Soc., vol. 257, n^\circ 1, 1980 , p. 39-71. MR 81a:57021 | Zbl 0423.57005 · Zbl 0423.57005 · doi:10.2307/1998124
[19] TURAEV (V. G.) . - Milnor invariants and Massey products . Zapiski Neuchnykh seminarov leningradskovo otdeleniya matematicheskovo instituta im. A.y. Stekhlor, Akad. Nauk SSSR, vol. 66, 1976 , p. 189-203. MR 56 #9538 | Zbl 0354.55013 · Zbl 0354.55013 · eudml:66800
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