Knot signature functions are independent
Authors:
Jae Choon Cha and Charles Livingston
Journal:
Proc. Amer. Math. Soc. 132 (2004), 28092816
MSC (2000):
Primary 57M25; Secondary 11E39
Published electronically:
April 21, 2004
MathSciNet review:
2054808
Fulltext PDF Free Access
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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 twosided 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 higherdimensional 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 305714, 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/S0002993904073782
PII:
S 00029939(04)073782
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
