Multivariable Alexander Invariants of Hypersurface Complements

On the Alexander invariants of hypersurface complements

Ja n 20 02 HYPERSURFACE COMPLEMENTS , ALEXANDER MODULES AND MONODROMY

ar X iv : h ep - t h / 93 09 02 9 v 1 6 S ep 1 99 3 Multivariable Invariants of Colored Links Generalizing the Alexander Polynomial ∗

We discuss multivariable invariants of colored links associated with the N dimensional root of unity representation of the quantum group. The invariants for N > 2 are generalizations of the multi-variable Alexander polynomial. The invariants vanish for disconnected links. We review the definition of the invariants through (1,1)-tangles. When (N, 3) = 1 and N is odd, the invariant does not vanis...

Invariants of Colored Links and a Property of the Clebsch-gordan Coefficients of U Q (g)

Abstract. We show that multivariable colored link invariants are derived from the roots of unity representations of Uq(g). We propose a property of the Clebsch-Gordan coefficients of Uq(g), which is important for defining the invariants of colored links. For Uq(sl2) we explicitly prove the property, and then construct invariants of colored links and colored ribbon graphs, which generalize the m...

Intersection Homology and Alexander Modules of Hypersurface Complements

Let V be a degree d, reduced hypersurface in CP, n ≥ 1, and fix a generic hyperplane, H. Denote by U the (affine) hypersurface complement, CP−V ∪H, and let U be the infinite cyclic covering of U corresponding to the kernel of the linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules Hi(U ;Q) of the hypersurface complement and show t...