Charles Livingston

TWISTED ALEXANDER INVARIANTS, REIDEMEISTER TORSION, AND CASSON—GORDON INVARIANTS

In this paper we extend several results about classical knot invariants derived from the infinite cyclic cover to the twisted case. Let X be a finite complex with fundamental group n, let o :nPG ̧(») be a linear representation where » is a finite dimensional vector space over a field F, and let e : nPZ be a homomorphism. Finally let X = be the infinite cyclic cover of X corresponding to e. The representation o restricts to give a representation of n 1 (X = ) to G ̧(») and one can define the twisted homology groups H i (X = ; »). These are F[Z] modules via the action of Z as the deck transformations of X = ; a polynomial representing the order of the F[Z] torsion of this module is called the twisted Alexander polynomial, * i , associated to the space and representations. In the case that X is a classical knot complement and o is a trivial one-dimensional representation, * 1 is the classical Alexander polynomial. We develop properties of these twisted Alexander modules and polynomials and prove the analogues of some of the classical results about the ordinary Alexander modules and polynomials; in particular we prove a number of results relating to their application in classical knot theory and concordance. One merit of this approach is that it gives a method of organizing non-abelian invariants of knots in a framework similar to the classical approach to abelian invariants. Of particular interest is that these invariants offer a 3-dimensional definition of certain Casson—Gordon invariants, they lead to an elementary proof that these Casson—Gordon invariants provide obstructions to knots being slice, and there are simple algorithms for their computation. We show that for certain representations of a cyclic cover of a knot complement an associated twisted polynomial must have a factorization of the form f (t) f (t~1) if the knot is slice. In this form the obstruction is seen as a direct generalization of the well-known result concerning the factorization of the standard Alexander polynomial of a slice knot. In a second paper [10] we will apply the results presented here to show that particular knots, e.g. 8 17 , are not concordant to their inverses and to show that positive mutation can change the concordance class of a knot, answering [12, 1.53].

Computations of the Ozsvath-Szabo knot concordance invariant

Ozsvath and Szabo have defined a knot concordance invariantthat bounds the 4-ball genus of a knot. Here we discuss shortcuts to its computation. We include examples of Alexander polynomial one knots for which the invariant is nontrivial, including all iterated untwisted positive doubles of knots with non- negative Thurston-Bennequin number, such as the trefoil, and explicit com- putations for several 10 crossing knots. We also note that a new proof of the Slice-Bennequin Inequality quickly follows from these techniques.

THE GASSNER REPRESENTATION FOR STRING LINKS

The Gassner representation of the pure braid group to

TWISTED KNOT POLYNOMIALS: INVERSION, MUTATION AND CONCORDANCE

GL_n(Z[Z^n])

A Survey of Classical Knot Concordance

can be extended to give a representation of the concordance group of

Ozsváth-Szabó and Rasmussen invariants of doubled knots

n

Algebraic and Heegaard–Floer invariants of knots with slice Bing doubles

-strand string links to

Vassiliev invariants of two component links and the Casson-Walker invariant

GL_n(F)

Group actions on fibered three-manifolds.

, where

HEEGAARD FLOER HOMOLOGY AND RATIONAL CUSPIDAL CURVES

F