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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A generalization of Torres' second relation


Author: Lorenzo Traldi
Journal: Trans. Amer. Math. Soc. 269 (1982), 593-610
MSC: Primary 57M25
MathSciNet review: 637712
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Abstract: Let $ L = {K_1} \cup \cdots \cup {K_\mu }$ be a tame link in $ {S^3}$ of $ \mu \geqslant 2$ components, and let $ {L_\mu }$ be its sublink $ {L_\mu } = L - {K_\mu }$. Let $ H$ and $ {H_\mu }$ be the abelianizations of $ {\pi _1}({S^3} - L)$ and $ {\pi _1}({S^3} - {L_\mu })$, respectively, and let $ {t_1}, \ldots ,{t_\mu }$ (resp., $ {t_1}, \ldots ,{t_{\mu - 1}}$) be the usual generators of $ H$ (resp., $ {H_\mu }$). If $ \phi :{\mathbf{Z}}H \to {\mathbf{Z}}{H_\mu }$ is the (unique) ring homomorphism with $ \phi ({t_i}) = {t_i}$ for $ 1 \leqslant i < \mu $, and $ \phi ({t_\mu }) = 1$, then Torres' second relation is equivalent to the statement that $ \phi {E_1}(L) = (({\prod _{i < \mu }}t_i^{{l_i}}) - 1) \cdot {E_1}({L_\mu })$, where for $ 1 \leqslant i < \mu $, $ {l_i}$ is the linking number $ {l_i} = l({K_i},\,{K_\mu })$. We prove that if $ I{H_\mu }$ is the augmentation ideal of $ {\mathbf{Z}}{H_\mu }$, then for any $ k \geqslant 2$,

$\displaystyle {E_{k - 1}}({L_\mu }) + \left( {\left( {\prod\limits_{i < \mu } {... ...q \phi {E_k}(L) \subseteq {E_{k - 1}}({L_\mu }) + I{H_\mu }\cdot{E_k}({L_\mu })$

and examples are given to indicate that either of these inclusions may be an equality. This theorem is used to generalize certain known properties of $ {E_1}$ to the higher ideals.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0637712-4
PII: S 0002-9947(1982)0637712-4
Keywords: Tame links, elementary ideals, Alexander polynomials, Torres' relations
Article copyright: © Copyright 1982 American Mathematical Society