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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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A generalization of Torres’ second relation
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by Lorenzo Traldi PDF
Trans. Amer. Math. Soc. 269 (1982), 593-610 Request permission

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 $k \geqslant 2$, \[ {E_{k - 1}}({L_\mu }) + \left ( {\left ( {\prod \limits _{i < \mu } {t_i^{{l_i}}} } \right ) - 1} \right )\cdot {E_k}({L_\mu }) \subseteq \phi {E_k}(L) \subseteq {E_{k - 1}}({L_\mu }) + I{H_\mu }\cdot {E_k}({L_\mu })\] 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.
References
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  • L. Traldi, On the determinantal ideals of link modules and a generalization of Torres’ second relation, Ph.D. Dissertation, Yale Univ., New Haven, Conn., 1980.
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 269 (1982), 593-610
  • MSC: Primary 57M25
  • DOI: https://doi.org/10.1090/S0002-9947-1982-0637712-4
  • MathSciNet review: 637712