Doubly semi-equivelar maps on the plane and the torus

Figures & data

Figure 1. Doubly semi-equivelar maps N_i’s of Types T_i’s: $i\in \{1,2,3,4\}.$

Figure 2. Doubly semi-equivelar maps N_i’s of Types T_i’s: $i\in \{5,8,11,12,13,18,19,20,21\}.$

Figure 3. Doubly semi-equivelar map of type T₁₆.

Figure 4. Doubly semi-equivelar maps N_i’s of types T_i’s, $i\in \{6,7,9,10,14,15,17\}.$

Figure 5. Doubly semi-equivelar maps of type T₂₂.

Figure 6. Path of type $P_{2i}$ (indicated by thick line).

Figure 7. Path of type $P_{2i-1}$ (indicated by thick line).

Figure 8. Paths of types $A_1,A_2,A_3.$

Figure 9. $M(i,j,k).$

Figure 10. Cylinder-I.

Figure 11. Cylinder-II.

Figure 12. $M(i,j,k).$

Table 1. DSEMs of type T₁ on the torus for $|V(M)|\le 18.$

$|V(M)|$	Isomorphic classes	Length of cycles	No of maps
9	$M(3,3,0),M(3,3,1)$	$(3,\{3,9\},3)$	2
	$M(3,3,2)$	$(3,\{9,9\},5)$
12	$M(4,3,0),M(4,3,2)$	$(4,\{3,6\},3)$	3
	$M(4,3,1)$	$(4,\{12,12\},4)$
	$M(4,3,3)$	$(4,\{12,12\},6)$
15	$M(5,3,0),M(5,3,3)$	$(5,\{3,15\},3)$	3
	$M(5,3,1),M(5,3,2)$	$(5,\{15,15\},4)$
	$M(5,3,4)$	$(5,\{15,15\},7)$
18	$M(6,3,0),M(6,3,4)$	$(6,\{3,9\},3)$	6
	$M(6,3,1),M(6,3,3)$	$(6,\{6,18\},4)$
	$M(6,3,2)$	$(6,\{9,9\},5)$
	$M(6,3,5)$	$(6,\{18,18\},8)$
	$M(3,6,0),M(3,6,2)$	$(3,\{6,18\},6)$
	$M(3,6,1)$	$(3,\{18,18\},7)$

Figure 13. Doubly semi-equivelar maps on the torus of type T₁.

Figure 14. Paths of types $B_1,B_2,B_3.$

Figure 15. $M(i,4,k)$ representation.

Table 2. DSEMs of type T₂ on the torus for $|V(M)|\le 24.$

$|V(M)|$	Isomorphic classes	Length of cycles	No. of maps
12	$M(3,4,0)$	$(3,\{4,4\},4)$	2
	$M(3,4,1),M(3,4,2)$	$(3,\{12,12\},5)$
16	$M(4,4,0),M(4,4,1)$	$(4,\{4,16\},4)$	2
	$M(4,4,2),M(4,4,3)$	$(4,\{8,16\},6)$
20	$M(5,4,0),M(5,4,2)$	$(5,\{4,20\},4)$	3
	$M(5,4,3),M(5,4,4)$	$(5,\{20,20\},7)$
	$M(5,4,1)$	$(5,\{20,20\},5)$
24	$M(6,4,0),M(6,4,3)$	$(6,\{4,8\},4)$	5
	$M(6,4,1),M(6,4,2)$	$(6,\{12,24\},5)$
	$M(6,4,4),M(6,4,5)$	$(6,\{12,24\},8)$
	$M(3,8,0)$	$(3,\{8,8\},8)$
	$M(3,8,1),M(3,8,2)$	$(3,\{24,24\},9)$

Figure 16. Paths of type $X_1,X_2,X_3.$

Figure 17. $M(i,3,k).$

Table 3. DSEMs of type T₃ on the torus for $|V(M)|\le 18.$

$|V(M)|$	Isomorphic classes	Length of cycles	No of maps
9	$M(3,3,0),M(3,3,2)$	$(3,\{3,9\},3)$	2
	$M(3,3,1)$	$(3,\{9,9\},4)$
12	$M(4,3,0),M(4,3,3)$	$(4,\{3,12\},3)$	2
	$M(4,3,1),M(4,3,2)$	$(4,\{6,12\},4)$
15	$M(5,3,0),M(5,3,4)$	$(5,\{3,15\},3)$	3
	$M(5,3,1),M(5,3,3)$	$(5,\{15,15\},4)$
	$M(5,3,2)$	$(5,\{15,15\},5)$
18	$M(6,3,0),M(6,3,5)$	$(6,\{3,18\},3)$	5
	$M(6,3,1),M(6,3,4)$	$(6,\{9,18\},4)$
	$M(6,3,2),M(6,3,3)$	$(6,\{6,9\},5)$
	$M(3,6,0),M(3,6,1)$	$(3,\{6,18\},6)$
	$M(3,6,2)$	$(3,\{18,18\},8)$

Figure 18. Paths of type $Y_1,Y_2,Y_3.$

Figure 19. $M(i,4,k)$ representation.

Table 4. DSEMs of type T₄ on the torus for $|V(M)|\le 24.$

$|V(M)|$	Isomorphic classes	Length of cycles	No of maps
12	$M(3,4,0),M(3,4,2)$	$(3,\{4,12\},4)$	2
	$M(3,4,1)$	$(3,\{12,12\},5)$
16	$M(4,4,0),M(4,3,3)$	$(4,\{4,16\},4)$	2
	$M(4,4,1),M(4,4,2)$	$(4,\{8,16\},5)$
20	$M(5,4,0),M(5,4,4)$	$(5,\{4,20\},4)$	3
	$M(5,4,1),M(5,4,3)$	$(5,\{20,20\},5)$
	$M(5,4,2)$	$(5,\{20,20\},6)$
24	$M(6,4,0),M(6,4,5)$	$(6,\{4,24\},4)$	5
	$M(6,4,1),M(6,4,4)$	$(6,\{12,24\},5)$
	$M(6,4,2),M(6,4,3)$	$(6,\{8,12\},6)$
	$M(3,8,0),M(3,8,1)$	$(3,\{8,24\},8)$
	$M(3,8,2)$	$(3,\{24,24\},10)$