Algebraic and Geometric Topology 4 (2004),
paper no. 19, pages 347-398.
Noncommutative knot theory
Tim D. Cochran
Abstract.
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.
Keywords.
Knot, Alexander module, Alexander polynomial, derived series, signature, Arf invariant
AMS subject classification.
Primary: 57M27.
Secondary: 20F14.
DOI: 10.2140/agt.2004.4.347
E-print: arXiv:math.GT/0206258
Submitted: 17 March 2004.
Accepted: 26 March 2004.
Published: 8 June 2004.
Notes on file formats
Tim D. Cochran
Department of Mathematics, Rice University
6100 Main Street, Houston, Texas 77005-1892, USA
Email: cochran@rice.edu
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