On the affine representations of the trefoil knot group

TL;DR

This paper classifies affine representations of the trefoil knot group in various geometries, explores their deformations, and investigates related affine isometry groups, revealing limitations in affine crystallographic groups in Minkowski space.

Contribution

It provides a complete classification of the trefoil knot group's affine representations and their geometric interpretations, including non-crystallographic affine groups.

Findings

Classified representations of the trefoil knot group in S^3 and SL(2,R).

Identified affine deformations and geometric interpretations.

Proved the non-existence of certain affine crystallographic groups in Minkowski space.

Abstract

The complete classification of representations of the Trefoil knot group G in S^{3} and SL(2,R), their affine deformations, and some geometric interpretations of the results, are given. Among other results, we also obtain the classification up to conjugacy of the non cyclic groups of affine Euclidean isometries generated by two isometries $\mu$ and $\nu$ such that $\mu^{2}=\nu^{3}=1$, in particular those which are crystallographic. We also prove that there are no affine crystallographic groups in the three dimensional Minkowski space which are quotients of G.