Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57M05, 57M25

Retrieve articles in all journals with MSC: 57M05, 57M25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0678352-1
Keywords: Tame links, linking numbers, elementary ideals, lower central series quotients, Alexander polynomials
Article copyright: © Copyright 1983 American Mathematical Society