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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Knot signature functions are independent


Authors: Jae Choon Cha and Charles Livingston
Journal: Proc. Amer. Math. Soc. 132 (2004), 2809-2816
MSC (2000): Primary 57M25; Secondary 11E39
Published electronically: April 21, 2004
MathSciNet review: 2054808
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Abstract | References | Similar Articles | Additional Information

Abstract: A Seifert matrix is a square integral matrix $V$ satisfying

\begin{displaymath}\det(V - V^T) =\pm 1. \end{displaymath}

To such a matrix and unit complex number $\omega$ there corresponds a signature,

\begin{displaymath}\sigma_\omega(V) = \mbox{sign}( (1 - \omega)V + (1 - \bar{\omega})V^T). \end{displaymath}

Let $S$ denote the set of unit complex numbers with positive imaginary part. We show that $\{\sigma_\omega\}_ { \omega \in S }$ is linearly independent, viewed as a set of functions on the set of all Seifert matrices.

If $V$ is metabolic, then $\sigma_\omega(V) = 0$ unless $\omega$ is a root of the Alexander polynomial, $\Delta_V(t) = \det(V - tV^T)$. Let $A$ denote the set of all unit roots of all Alexander polynomials with positive imaginary part. We show that $\{\sigma_\omega\}_ { \omega \in A }$ is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices.

To each knot $K \subset S^3$ one can associate a Seifert matrix $V_K$, and $\sigma_\omega(V_K)$ induces a knot invariant. Topological applications of our results include a proof that the set of functions $\{\sigma_\omega\}_ { \omega \in S }$ is linearly independent on the set of all knots and that the set of two-sided averaged signature functions, $\{\sigma^*_\omega\}_ { \omega \in S }$, forms a linearly independent set of homomorphisms on the knot concordance group. Also, if $\nu \in S$ is the root of some Alexander polynomial, then there is a slice knot $K$ whose signature function $\sigma_\omega(K)$ is nontrivial only at $\omega = \nu$and $\omega = \overline{\nu}$. We demonstrate that the results extend to the higher-dimensional setting.


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Additional Information

Jae Choon Cha
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Address at time of publication: Information and Communications University, Daejeon 305-714, Republic of Korea
Email: jccha@indiana.edu, jccha@icu.ac.kr

Charles Livingston
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: livingst@indiana.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07378-2
PII: S 0002-9939(04)07378-2
Keywords: Knot, signature, metabolic forms, concordance
Received by editor(s): January 29, 2003
Received by editor(s) in revised form: June 12, 2003
Published electronically: April 21, 2004
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2004 American Mathematical Society