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
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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?)

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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-1012525-7
Article copyright: © Copyright 1990 American Mathematical Society