Multiple k-isohedral tilings beyond Type 6

Determine whether convex pentagonal monotiles belonging to the Type 7–13 families and not contained in T_1 or T_2 can generate multiple types of k-isohedral tilings.

Background

The paper observes that monotiles belonging only to Type 6 can generate both 2-isohedral and 4-isohedral tilings. It then conjectures that analogous behavior is unlikely for monotiles belonging only to Types 7–13, largely because their representative tilings require reflected tiles. This remains an explicitly stated conjectural issue.

References

Based on these observations, we conjecture that it is unlikely that convex pentagonal monotiles belonging to the Type 7--13 families, which are not contained in $T_{1}$ or $T_{2}$, can generate multiple types of $k$-isohedral tilings.

— Convex pentagonal monotiles in the 15 Type families  (2501.07090 - Sugimoto, 13 Jan 2025) in Appendix D