# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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by Lorenzo Traldi
Trans. Amer. Math. Soc. 275 (1983), 309-318 Request permission

## 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 $0 < k < \mu$ $\sum \limits _{i = 0}^{k - 1} {{E_{\mu - k + i}}(L) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}} = \sum \limits _{i = 0}^{k - 1} {{E_{\mu - k + i}}(\mathcal {L}) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}}} }$ 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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