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
, 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,
, these invariants
reflect the structure of
as a module over
(here
is the
term of the derived series of G). Hence any phenomenon associated to
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
higher-order Alexandermodule, , considered
as a –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
