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
- Richard H. Crowell and Ralph H. Fox, Introduction to knot theory, Graduate Texts in Mathematics, No. 57, Springer-Verlag, New York-Heidelberg, 1977. Reprint of the 1963 original. MR 0445489
- Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215
- Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
- H. Seifert, Über das Geschlecht von Knoten, Math. Ann. 110 (1935), no. 1, 571–592 (German). MR 1512955, DOI 10.1007/BF01448044
- Guillermo Torres, On the Alexander polynomial, Ann. of Math. (2) 57 (1953), 57–89. MR 52104, DOI 10.2307/1969726 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.
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