Holonomy preserving transformations of weighted graphs and its application to knot theory

Abstract

Goda showed that the twisted Alexander polynomial can be recovered from the zeta function of a matrix-weighted graph. Motivated by this, we study transformations of weighted graphs that preserve this zeta function, introducing a notion of holonomy as an analogy for the accumulation of weights along cycles. We extend the framework from matrices to group elements, and show that holonomy preserving transformations correspond to transformations of group presentations and preserve the twisted Alexander polynomial from a graph-theoretic viewpoint. We also generalize to quandle-related structures, where the holonomy condition coincides with the Alexander pair condition of Ishii and Oshiro. This perspective allows us to view knot diagrams as covering-like structures enriched with holonomy.