Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Milnor's invariants and the completions of link modules


Author: Lorenzo Traldi
Journal: Trans. Amer. Math. Soc. 284 (1984), 401-424
MSC: Primary 57M05; Secondary 57M25
DOI: https://doi.org/10.1090/S0002-9947-1984-0742432-3
MathSciNet review: 742432
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ L$ be a tame link of $ \mu \geqslant 2$ components in $ {S^3}$, $ H$ the abelianization of its group $ {\pi _1}({S^3} - L)$, and $ IH$ the augmentation ideal of the integral group ring $ {\mathbf{Z}}H$. The $ IH$-adic completions of the Alexander module and Alexander invariant of $ L$ are shown to possess presentation matrices whose entries are given in terms of certain integers $ \mu ({i_1}, \ldots ,{i_q})$ introduced by J. Milnor. Various applications to the theory of the elementary ideals of these modules are given, including a condition on the Alexander polynomial necessary for the linking numbers of the components of $ L$ with each other to all be zero. In the special case $ \mu = 2$, it is shown that the various Milnor invariants $ \bar \mu ([r + 1,s + 1])$ are determined (up to sign) by the Alexander polynomial of $ L$, and that this Alexander polynomial is 0 iff $ \bar \mu ([r + 1,s + 1]) = 0$ for all $ r,s \geqslant 0$ with $ r + s$ even; also, the Chen groups of $ L$ are determined (up to isomorphism) by those nonzero $ \bar \mu ([r + 1,s + 1])$ with $ r + s$ minimal. In contrast, it is shown by example that for $ \mu \geqslant 3$ the Alexander polynomials of a link and its sublinks do not determine its Chen groups.


References [Enhancements On Off] (What's this?)

  • [1] N. Bourbaki, Commutative algebra, Hermann, Paris and Addison-Wesley, Reading, Mass., 1972. MR 0360549 (50:12997)
  • [2] K.-T. Chen, R. H. Fox and R. C. Lyndon, Free differential calculus. IV, Ann. of Math. (2) 68 (1958), 81-95. MR 0102539 (21:1330)
  • [3] J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra, Pergamon Press, Oxford, 1969, pp. 329-358. MR 0258014 (41:2661)
  • [4] R. H. Crowell, The derived module of a homomorphism, Adv. in Math. 6 (1971), 210-238. MR 0276308 (43:2055)
  • [5] -, Torsion in link modules, J. Math. Mech. 14 (1965), 289-298. MR 0174606 (30:4807)
  • [6] R. H. Crowell and D. Strauss, On the elementany ideals of link modules, Trans. Amer. Math. Soc. 142 (1969), 93-109. MR 0247625 (40:889)
  • [7] R. H. Fox, Free differential calculus. I, Ann. of Math. (2) 57 (1953), 547-560. MR 0053938 (14:843d)
  • [8] -, Free differential calculus. II, Ann. of Math. (2) 59 (1954), 196-210. MR 0062125 (15:931e)
  • [9] J. A. Hillman, Alexander ideals of links, Springer-Verlag, Berlin and New York, 1981. MR 653808 (84j:57004)
  • [10] N. Jacobson, Basic algebra I, Freeman, San Francisco, Calif., 1974. MR 0356989 (50:9457)
  • [11] M. E. Kidwell, On the Alexander polynomials of certain three-component links, Proc. Amer. Math. Soc. 71 (1978), 351-354. MR 0482737 (58:2791)
  • [12] W. S. Massey, Completion of link modules, Duke Math. J. 47 (1980), 399-420. MR 575904 (81g:57004)
  • [13] J. Milnor, Link groups, Ann. of Math. (2) 59 (1954), 177-195. MR 0071020 (17:70e)
  • [14] -, Isotopy of links, Algebraic Geometry and Topology, Princeton Univ. Press, Princeton, N. J., 1957, pp. 280-306. MR 0092150 (19:1070c)
  • [15] K. Murasugi, On Milnor's invariant for links, Trans. Amer. Math. Soc. 124 (1966), 94-110. MR 0198453 (33:6611)
  • [16] -, On Milnor's invariant for links. II. The Chen group, Trans. Amer. Math. Soc. 148 (1970), 41-61. MR 0259890 (41:4519)
  • [17] D. G. Northcott, Finite free resolutions, Cambridge Univ. Press, Cambridge, 1976. MR 0460383 (57:377)
  • [18] D. Rolfsen, Knots and links, Publish or Perish, Berkeley, Calif., 1976. MR 0515288 (58:24236)
  • [19] N. Smythe, Isotopy invariants of links and the Alexander matrix, Amer. J. Math. 89 (1967), 693-704. MR 0219056 (36:2139)
  • [20] G. Torres, On the Alexander polynomial, Ann. of Math. (2) 57 (1953), 57-89. MR 0052104 (14:574a)
  • [21] L. Traldi, The determinantal ideals of link modules. I, Pacific J. Math. 101 (1982), 215-222. MR 671854 (84h:57004)
  • [22] -, Linking numbers and the elementary ideals of links, Trans. Amer. Math. Soc. 275 (1983), 309-318. MR 678352 (84d:57002)
  • [23] -, Some properties of the determinantal ideals of link modules, Kobe J. Math, (to appear).
  • [24] O. Zariski and P. Samuel, Commutative algebra, Vol. II, Van Nostrand, Princeton, N. J., 1960. MR 0120249 (22:11006)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57M05, 57M25

Retrieve articles in all journals with MSC: 57M05, 57M25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0742432-3
Keywords: Tame links, Milnor's invariants, $ I$-adic completions, Chen groups, elementary ideals, Alexander polynomials
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society