4-Legendrian-Racks

Program for classifying 4-Legendrian racks up to isomorphism. Data for those up to order 6.

Based on the online rack library of Petr Vojtěchovský and Seung Yeop Yang.

This repo accompanies the paper "Distinguishing power of 4-Legendrian permutation racks" [arXiv:2510.26619], which is joint work with Peyton Phinehas Wood.

Usage

4-legendrian-finder.txt exhaustively searches for 4-Legendrian structures on all racks of a given order n in Vojtěchovský and Yang's library, searches for isomorphisms between them, and outputs a list of all 4-Legendrian racks of order n up to isomorphism.

For example, to run the search for all n between 1 and 3, load Vojtěchovský and Yang's library and then run the following. The following assumes that you've saved 4-legendrian-finder.txt in the same folder as the rack library; if not, then replace "LRQ.path" below and in 4-legendrian-finder.txt with the path to wherever you've saved 4-legendrian-finder.txt.

for n in [1..3] do
	ReadAsFunction(Concatenation(LRQ.path, "4-legendrian_finder.txt"))()(n);
od;

Note

In the program and data, 4-Legendrian racks are stored as lists of the form (R, uL, uR, dR, dL), where R is the underlying rack. This ordering is distinct from the usual ordering.