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

DOI:
https://doi.org/10.1090/S0002-9939-04-07378-2

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 satisfying

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

Let denote the set of unit complex numbers with positive imaginary part. We show that is linearly independent, viewed as a set of functions on the set of all Seifert matrices.

If is metabolic, then unless is a root of the Alexander polynomial, . Let denote the set of all unit roots of all Alexander polynomials with positive imaginary part. We show that is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices.

To each knot one can associate a Seifert matrix , and induces a knot invariant. Topological applications of our results include a proof that the set of functions is linearly independent on the set of all knots and that the set of two-sided averaged signature functions, , forms a linearly independent set of homomorphisms on the knot concordance group. Also, if is the root of some Alexander polynomial, then there is a slice knot whose signature function is nontrivial only at and . 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:
https://doi.org/10.1090/S0002-9939-04-07378-2

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