Noncommutative knot theory

The classical abelian invariants of a knot are the Alexander module, which is the first homology group of the the unique infinite cyclic covering space of $S^{3} - K$ , considered as a module over the (commutative) Laurent polynomial ring, and the Blanchfield linking pairing defined on this module. From the perspective of the knot group, $G$ , these invariants reflect the structure of $G^{(1)} ∕ G^{(2)}$ as a module over $G ∕ G^{(1)}$ (here $G^{(n)}$ is the $n^{th}$ term of the derived series of G). Hence any phenomenon associated to $G^{(2)}$ is invisible to abelian invariants. This paper begins the systematic study of invariants associated to solvable covering spaces of knot exteriors, in particular the study of what we call the $n^{th}$ higher-order Alexander module, $G^{(n + 1)} ∕ G^{(n + 2)}$ , considered as a $ℤ [G ∕ G^{(n + 1)}]$ –module. We show that these modules share almost all of the properties of the classical Alexander module. They are torsion modules with higher-order Alexander polynomials whose degrees give lower bounds for the knot genus. The modules have presentation matrices derived either from a group presentation or from a Seifert surface. They admit higher-order linking forms exhibiting self-duality. There are applications to estimating knot genus and to detecting fibered, prime and alternating knots. There are also surprising applications to detecting symplectic structures on 4–manifolds. These modules are similar to but different from those considered by the author, Kent Orr and Peter Teichner and are special cases of the modules considered subsequently by Shelly Harvey for arbitrary 3–manifolds.

PDF Access Denied

Warning: We have not been able to recognize your IP address 47.88.87.18 as that of a subscriber to this journal. Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form.

Or, visit our subscription page for instructions on purchasing a subscription. You may also contact us at contact@msp.org or by using our contact form.

Or, you may purchase this single article for USD 29.95:

Keywords

knot, Alexander module, Alexander polynomial, derived series, signature, Arf invariant

References

Forward citations

Publication

Received: 17 March 2004

Accepted: 26 March 2004

Published: 8 June 2004

Authors

Tim D Cochran
Department of Mathematics Rice University 6100 Main Street Houston TX 77005-1892 USA

Volume 4, issue 1 (2004)

Tim D Cochran

Abstract