Noncommutative knot theory

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 $G^{\left(1\right)}∕G^{\left(2\right)}$ as a module over $G∕G^{\left(1\right)}$ (here $G^{\left(n\right)}$ is the $n^{th}$ term of the derived series of G). Hence any phenomenon associated to $G^{\left(2\right)}$ 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, $G^{\left(n+1\right)}∕G^{\left(n+2\right)}$, considered as a $\mathbb{Z}\left[G∕G^{\left(n+1\right)}\right]$–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.