The Gassner Representation for String Links

The Gassner representation of the pure braid group to GLn(Z[Z ]) can be extended to give a representation of the concordance group of n-strand string links to GLn(F ), where F is the field of quotients of Z[Z ], F = Q(t1, · · · , tn); this was first observed by Le Dimet. Here we give a cohomological interpretation of this extension. Our first application is to prove that the representation is hermitian, extending a known result for braids. A simple proof of the concordance invariance of the represenation also follows. The cohomological approach leads to algorithms for computing the Gassner representation, and these in turn yield connections between the Gassner represention of a string link and the Alexander matrix of the link closure of that string link; a new factorization result for Alexander polynomials follows. A random walk, or probablistic, approach to the Gassner representation is given, extending previous work concerning the 1-variable Burau representation. We next show that by suitably normalizing, the Gassner matrix determines, and is determined by, finite type link invariants. The paper concludes with an interpretation of the determinant of the Gassner matrix in terms of Reidemeister torsion, yielding an alternative approach to the factorization of the Alexander polynomial.