Periodic orbits of a dynamical system related to a knot

TL;DR

This paper studies the structure of a graph representing representations of a knot group’s commutator subgroup into Z/p, revealing the cycle lengths which relate to the underlying knot's properties.

Contribution

It provides a detailed description of the cycle lengths in the graph associated with these representations, extending previous work by Silver and Williams.

Findings

Cycle lengths in the graph are characterized explicitly.

The structure of the representation space is linked to the graph's cycle properties.

Results enhance understanding of knot group representations into finite cyclic groups.

Abstract

We consider the space of all representations of the commutator subgroup of a knot group into Z/p, p is prime. As proven by D. Silver and S. Williams, this space can be completely described by a finite oriented graph. We describe the lengths of cycles in this graph.