Holonomy preserving transformations of weighted graphs and its application to knot theory
Atsuhide Nagasaka
TL;DR
The paper develops holonomy-preserving transformations for weighted graphs to study the invariance of the twisted Alexander polynomial, strengthening Goda’s link between twisted invariants and matrix-weighted zeta functions. It extends the framework from matrix weights to group elements and further to quandles, establishing that holonomy-preserving transformations correspond to strong, base-point preserving equivalences of group presentations and preserve the twisted Alexander polynomial from a graph-theoretic perspective. By introducing group-weighted graphs, free differential calculus, and explicit elementary transformations, the authors provide a graph-theoretic proof of invariance and connect to Alexander pairs and quandles (Ishii-Oshiro). The work offers a unifying viewpoint where knot diagrams are interpreted as holonomy-enriched coverings, enabling invariant construction under Reidemeister moves and enriching the algebraic toolkit for knot invariants with graph-theoretic and quandle-theoretic methods.
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.