TWISTED ALEXANDER POLYNOMIAL AND MATRIX-WEIGHTED ZETA FUNCTION

摘要

The twisted Alexander polynomial is an invariant of the pair of a knot and its group representation. Herein, we introduce a digraph obtained from an oriented knot diagram, which is used to study the twisted Alexander polynomial of knots. In this context, we show that the inverse of the twisted Alexander polynomial of a knot may be regarded as the matrix-weighted zeta function that is a generalization of the Ihara-Selberg zeta function of a directed weighted graph.