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Transactions of the American Mathematical Society

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



Linking numbers and the elementary ideals of links

Author: Lorenzo Traldi
Journal: Trans. Amer. Math. Soc. 275 (1983), 309-318
MSC: Primary 57M05; Secondary 57M25
MathSciNet review: 678352
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Abstract: Let $ L = {K_1} \cup \, \cdots \cup {K_\mu } \subseteq {S^3}$ be a tame link of $ \mu \geqslant 2$ components, and $ H$ the abelianization of $ G = {\pi _1}({S^3} - L)$. Let $ \mathcal{L} = ({\mathcal{L}_{ij}})$ be the $ \mu \times \mu $ matrix with entries in $ \mathbf{Z}H$ given by $ \mathcal{L}{_{ii}} = \sum\nolimits_{k \ne i} {l({K_i},{K_k}) \cdot ({t_k} - 1)} $ and for $ i \ne j\,{\mathcal{L}_{ij}} = l({K_i},{K_j}) \cdot (1 - {t_i})$. Then if $ 0 < k < \mu $

$\displaystyle \sum\limits_{i = 0}^{k - 1} {{E_{\mu - k + i}}(L) \cdot {{(IH)}^{... ...{k - 1} {{E_{\mu - k + i}}(\mathcal{L}) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}}} } $

Various consequences of this equality are derived, including its application to the reduced elementary ideals. These results are used to give several different characterizations of links in which all the linking numbers are zero.

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Keywords: Tame links, linking numbers, elementary ideals, lower central series quotients, Alexander polynomials
Article copyright: © Copyright 1983 American Mathematical Society

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