Pauli-Lubanski scalar in the Polygon Approach to 2+1-Dimensional Gravity

In this paper we derive an expression for the conserved Pauli-Lubanski scalar in 't Hooft’s polygon approach to 2+1-dimensional gravity coupled to point particles. We find that it is represented by an extra spatial shift $\mathrm{\Delta }$ in addition to the usual identification rule (being a rotation over the cut). For two particles this invariant is expressed in terms of 't Hooft’s phase-space variables and we check its classical limit.