Module d'Alexander et repr\'esentations m\'etab\'eliennes

TL;DR

This paper generalizes the detection of Alexander polynomial roots via knot group representations by extending from 2x2 to nxn upper triangular matrices, revealing the Alexander module decomposition over complex numbers.

Contribution

It introduces a new approach using higher-dimensional upper triangular matrix representations to analyze the Alexander module of knots.

Findings

Generalization of Alexander polynomial root detection

Decomposition of Alexander module with complex coefficients

Extension from 2x2 to nxn matrix representations

Abstract

It is known, since works of Burde and de Rham, that one can detect the roots of the Alexander polynomial of a knot by the study of the representations of the knot group into the group of the invertible upper triangular $2x2$ matrices. In this work, we propose to generalize this result by considering the representations of the knot group into the group of the invertible upper triangular $nxn$ matrices, $n\geq 2$. This approach will enable us to find the decomposition of the Alexander module with complex coefficients.