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)



The Arf and Sato link concordance invariants

Author: Rachel Sturm Beiss
Journal: Trans. Amer. Math. Soc. 322 (1990), 479-491
MSC: Primary 57M25
MathSciNet review: 1012525
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Kervaire-Arf invariant is a $ Z/2$ valued concordance invariant of knots and proper links. The $ \beta $ invariant (or Sato's invariant) is a $ Z$ valued concordance invariant of two component links of linking number zero discovered by J. Levine and studied by Sato, Cochran, and Daniel Ruberman. Cochran has found a sequence of invariants $ \{ {\beta _i}\} $ associated with a two component link of linking number zero where each $ {\beta _i}$ is a $ Z$ valued concordance invariant and $ {\beta _0} = \beta $. In this paper we demonstrate a formula for the Arf invariant of a two component link $ L = X \cup Y$ of linking number zero in terms of the $ \beta $ invariant of the link:

$\displaystyle \operatorname{arf} (X \cup Y) = \operatorname{arf} (X) + \operatorname{arf} (Y) + \beta (X \cup Y)\quad (\bmod 2).$

This leads to the result that the Arf invariant of a link of linking number zero is independent of the orientation of the link's components. We then establish a formula for $ \vert\beta \vert$ in terms of the link's Alexander polynomial $ \Delta (x,y) = (x - 1)(y - 1)f(x,y)$:

$\displaystyle \vert\beta (L)\vert = \vert f(1,1)\vert.$

Finally we find a relationship between the $ {\beta _i}$ invariants and linking numbers of lifts of $ X$ and $ Y$ in a $ Z/2$ cover of the compliment of $ X \cup Y$.

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

  • [A] C. Arf, Untersuchungen über quadratische Formen in Körpern der Charackteristik $ 2$, Crelles Math. J. 183 (1941). MR 0008069 (4:237f)
  • [C] T. Cochran, Concordance invariants of coefficients of Conway's link polynomial, Invent. Math. 82 (1985), 527-41. MR 811549 (87c:57002)
  • [Co] D. Cooper, The universal Abelian cover of a link (R. Brown and T. L. Thickstun, eds.), Lecture Notes Ser., vol. 48, London Math. Soc., 1979.
  • [KM] M. Kervaire and J. Milnor, Groups of homotopy spheres, Ann. of Math. 77 (1963). MR 0148075 (26:5584)
  • [L] J. Levine, Polynomial invariants of knots of codimension two, Ann. of Math. 84 (1966). MR 0200922 (34:808)
  • [M] K. Murasugi, On the Arf invariant of links, preprint. MR 727081 (85j:57007)
  • [R] R. Robertello, An invariant of knot cobordism, Comm. Pure Appl. Math. 18 (1965). MR 0182965 (32:447)
  • [S] R. Sato, Corbodisms of semi-boundary links, preprint.

Similar Articles

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

Retrieve articles in all journals with MSC: 57M25

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society