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Journal of the London Mathematical Society Advance Access originally published online on August 26, 2007
Journal of the London Mathematical Society 2007 76(1):105-121; doi:10.1112/jlms/jdm004
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© 2007 London Mathematical Society

Twisted Alexander polynomials of plane algebraic curves

J. I. Cogolludo Agustín

Departamento de Matemáticas
Universidad de Zaragoza
C. Pedro Cerbuna 12
50009 Zaragoza
Spain

V. Florens

Departamento de Álgebra y Geometría
Universidad de Valladolid
Prado de la Magdalena
47005 Valladolid
Spain
vincent_florens{at}yahoo.fr

In the present paper, Alexander polynomials of plane algebraic curves twisted by linear representations are considered. They are shown to divide the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of the Blanchfield intersection form. Specializing to the classical case, this gives a divisibility formula in the sense of Libgober's divisibility theorem. Examples of twisted polynomials for some algebraic curves are explicitly calculated showing that they can detect Zariski pairs of equivalent Alexander polynomials and that they are sensitive to nodal degenerations.

jicogo{at}unizar.es

2000 Mathematics Subject Classification 57M05, 57Q10, 58K65, 14H30, 14B05, 55N33.

The first author was partially supported by MTM2004-08080-C02-02. The second author was supported by sMCHF-2001-0615.

Received May 18, 2005; revised August 1, 2006; published online August 26, 2007.


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