Noncommutative knot theory.

By Tim D. Cochran. 2002

125 citations recorded by the source corpus.

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 t h 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 t h higher-order Alexander module, G ( + 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e689d8174c2f0256d1de973a8b91fdbafbc8d7fc

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Citations in the corpus

The Thurston norm, fibered manifolds and twisted Alexander polynomials (2005, background): “This should be compared to the situation of [Co04] (cf. also [Ha06]) : Cochran’s sequence of higher order Alexander polynomials gives a never decreasing sequence of lower bounds on the genus of a knot.”
The Baum-Connes and the Farrell-Jones Conjectures in K- and L-Theory (2004, result): “We also refer to the survey article [68] about non-commutative geometry and knot theory.”
Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm (2002, background): “The result when X is a knot complement appears in [3] although it uses some of our work.”
Primary decomposition and the fractal nature of knot concordance (2009, background): “To be more specific, recall that the higher-order Alexander modules of a knot have a purely group theoretic description as the quotient, G(i)/G(i+1), of successive terms in the derived series of the fundamental group of the zero framed surgery on the knot [10,18].”
Structure in the classical knot concordance group (2002, background): “In fact it is proved in [C] that the derived series of a knot group cannot stabilize unless it has Alexander polynomial 1 and that of π1(M) cannot stabilize unless the Alexander polynomial has degree 0 or 2.”
Homology cobordism invariants and the Cochran-Orr-Teichner filtration of the link concordance group (2006, background): “Proceeding in this way, defining G r to be the radical of [G (n−1) r , G (n−1) r ], leads to what has been called the rational derived series of G [20] [8] [13].”
Knot concordance and von Neumann $\rho$-invariants (2004, methodology): “Theorem 2.9. There is a constant C, depending only on the local geometry of M, such that for all -covers of M | (M)| C · volume(M)”
Knot concordance and higher-order Blanchfield duality (2007, background): “Such modules were named higher-order Alexander modules in [7; 12; 23].”
The structure of the rational concordance group of knots (2006, methodology): “It is known that G (n) Q is a normal subgroup of G [11, 19].”
A SURVEY OF TWISTED ALEXANDER POLYNOMIALS (2009, methodology): “variants corresponding to ﬁnite dimensional representations over non–commutative rings. The study of such invariants (often referred to as higher order Alexander polynomials) was initiated by Cochran [Co04], building on ideas of Cochran, Orr and Teichner [COT03]. The notion of higher order Alexander”