Determine whether Types 7–13 outside Types 1 and 2 admit multiple transitivity classes

Determine whether every convex pentagonal monotile belonging only to one of the Type 6–13 families can generate tiling patterns other than its family’s representative tiling, and use this determination to test the conjecture that Type 7–13 monotiles outside Types 1 and 2 cannot generate multiple types of k-isohedral tilings.

Background

The paper observes that Type 6 monotiles outside Type 1 can generate both 2-isohedral and 4-isohedral tilings. It conjectures that analogous Type 7–13 monotiles outside Types 1 and 2 cannot generate multiple types of k-isohedral tilings. Verification requires classifying whether monotiles belonging only to each Type 6–13 family admit tiling patterns beyond the corresponding representative pattern, with particular emphasis on Type 6 as a possible source of exceptions.