Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-04-07378-2
Published electronically: April 21, 2004
MathSciNet review: 2054808
Full-text PDF

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.


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

  • 1. A. Casson and C. Gordon, Cobordism of classical knots, A la recherche de la topologie perdue, ed. by Guillou and Marin, Progress in Mathematics, Volume 62, 1986. (Originally published as Orsay Preprint, 1975.) MR 88k:57002
  • 2. A. Casson and C. Gordon, On slice knots in dimension three, Algebraic and Geometric Topology, Proc. Sympos. Pure Math., XXXII, Part 2, Amer. Math. Soc., Providence, RI, 1978, pp. 39-53. MR 81g:57003
  • 3. T. Cochran, K. Orr, and P. Teichner, Knot concordance, Whitney towers and $L^2$ signatures, Annals of Math. (2) 157 (2003), 433-519.
  • 4. P. Gilmer, Slice knots in $S\sp{3}$, Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 135, 305-322. MR 85d:57004
  • 5. P. Gilmer, Classical knot and link concordance, Comment. Math. Helv. 68 (1993), 1-19. MR 94c:57007
  • 6. T. Kim, Obstructions to slicing and doubly slicing knots, Thesis, Indiana University, May 2003.
  • 7. J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229-244. MR 39:7618
  • 8. J. Levine, Invariants of knot cobordism, Invent. Math. 8 (1969), 98-110. MR 40:6563
  • 9. J. Levine, Metabolic and hyperbolic forms from knot theory, J. Pure Appl. Algebra 58 (1989), 251-260. MR 90h:57027
  • 10. H. Seifert, Über das Geschlecht von Knoten, Math. Ann. 110 (1934), 571-592.
  • 11. N. Stoltzfus, Unraveling the integral knot concordance group, Memoirs Amer. Math. Soc. 12 (1977), no. 192. MR 57:7616
  • 12. D. W. Sumners, Invertible knot cobordisms, Comment. Math. Helv. 46 (1971), 240-256. MR 44:7535
  • 13. A. Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969), 251-264. MR 40:2104

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57M25, 11E39

Retrieve articles in all journals with MSC (2000): 57M25, 11E39


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

American Mathematical Society