Wieland gyration for triangular fully packed loop configurations

TL;DR

This paper extends Wieland gyration to triangular fully packed loop configurations, demonstrating its eventual stabilization and providing new insights and proofs for properties of TFPLs.

Contribution

It introduces Wieland gyration for TFPLs, analyzes its behavior, and offers new proofs of key properties, advancing understanding of TFPL symmetry and invariants.

Findings

Repeated gyration leads to stable configurations.

Characterization of invariant configurations.

Simplified proofs of TFPL properties.

Abstract

Triangular fully packed loop configurations (TFPLs) emerged as auxiliary objects in the study of fully packed loop configurations on a square (FPLs) corresponding to link patterns with a large number of nested arches. Wieland gyration, on the other hand, was invented to show the rotational invariance of the numbers $A_\pi$ of FPLs corresponding to a given link pattern $\pi$. The focus of this article is the definition and study of Wieland gyration on TFPLs. We show that the repeated application of this gyration eventually leads to a configuration that is left invariant. We also provide a characterization of such stable configurations. Finally we apply our gyration to the study of TFPL configurations, in particular giving new and simple proofs of several results.