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 ofterm of the derived series of G). Hence any phenomenon associated to Gis 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,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.

Link concordance, homology cobordism, and Hirzebruch-type defects from iterated $p$-covers

“…Roughly speaking, L 2 -signatures of the infected manifold or link associated to certain solvable coefficient systems reflect L 2 -signatures of the infection knot K associated to much simpler coefficient systems (e.g., abelian or metabelian ones). All recent works mentioned above [19,20,21,13,18,30,31,32,26,16,17,9] depend on results of this type. When the infection knot K is torsion, however, those L 2 -signatures have failed to detect anything.…”

“…(For definitions of concordance and the Cochran-Orr-Teichner filtration, see Sections 6 and 8.) There are several recent related results using infection, including works of Cochran, Taehee Kim, Harvey, Leidy, and the author [13,18,30,31,32,26,16,17,9]. Such reams of results lead us to study how to detect the effect of infection along a curve contained in a higher term of the derived series of the fundamental group, up to concordance and homology cobordism.…”

Structure in the classical knot concordance group

“…A DNA strand is a million times longer than the diameter of a cell and topoisomerases miraculously find the precise locations on this strand to perform their genetic modifications. Analogously, the knot group is infinitely long as measured by the derived series [C,Corollary 4.8] and we use extremely precise control of this when choosing the axis (especially in [CT]). On the other hand, the infection knots J that we use in the modification are quite robust just like a virus seems to be a very robust thing.…”

Link concordance and generalized doubling operators

“…The image h.8 9 / is the mirror image, x 8 9 , of 8 9 and the image of 1 is a ribbon disk, 2 for x 8 9 . Since 8 9 is isotopic to its mirror image, this can be viewed as another ribbon disk for 8 9 . The kernel of the map .i 2 / , as above, is h .…”