[thick] (1.1, 0) – (1.9,0) ;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (3.1, 0.07) – (3.9,0.07) ;
[thick] (3.1, -0.07) – (3.9,-0.07) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (3.4,0) – (3.6,0.2);
[thick] (3.4,0) – (3.6,-0.2);

24(c-a) = 6 N + 2 ,     4(2a-c) = 12N^2-18N+6 ,

Δ= {(6k,6k+2)_k=1,…,N-2,6N-6,2N } ,     𝔤 =  (𝔤_2)_4N⊕(𝔰𝔲_2)_16 .

§.§.§ $\mathcal S^{(N)}_{D_4,3}(1,0,1)$

[fill= white] (0,0) circle [radius=0.1] node[below] 1;
[fill= white] (1,0) circle [radius=0.1] node[below] 2;
[fill= white] (2,0) circle [radius=0.1] node[below] N+1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N+1;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N+1;
[thick] (0.1, 0) – (0.9,0) ;
[thick] (1.1, 0) – (1.9,0) ;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (3.1, 0.07) – (3.9,0.07) ;
[thick] (3.1, -0.07) – (3.9,-0.07) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (3.4,0) – (3.6,0.2);
[thick] (3.4,0) – (3.6,-0.2);

24(c-a) = 6 N + 5 ,     4(2a-c) = 12N^2 +2N-3 ,

Δ= {(6k,6k+2)_k=1,…,N-1,6N} ,     𝔤 =  (𝔰𝔲_2)_12N+2⊕(𝔰𝔲_2)_16 .

§.§.§ $\mathcal R^{(N)}_{D_4,3}(0,0,2)$

[fill= white] (0,0) circle [radius=0.1] node[below] 1;
[fill= white] (1,0) circle [radius=0.1] node[below] 2;
[fill= white] (2,0) circle [radius=0.1] node[below] N+2;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N+2;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N+2;
[thick] (0.1, 0) – (0.9,0) ;
[thick] (1.1, 0) – (1.9,0) ;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (3.1, 0.07) – (3.9,0.07) ;
[thick] (3.1, -0.07) – (3.9,-0.07) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (3.4,0) – (3.6,0.2);
[thick] (3.4,0) – (3.6,-0.2);

24(c-a) = 6 N + 8 ,     4(2a-c) = 12N^2 +14N ,

Δ= {(6k,6k+2)_k=1,…,N} ,     𝔤 =  (𝔰𝔲_3)_12N+4⊕(𝔰𝔲_2)_16 .

§.§.§ $\mathcal T^{(N)}_{D_4,3}(0,1,0)$

[fill= white] (0,0) circle [radius=0.1] node[below] 1;
[fill= white] (1,0) circle [radius=0.1] node[below] 2;
[fill= white] (2,0) circle [radius=0.1] node[below] N+1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N;
[thick] (0.1, 0) – (0.9,0) ;
[thick] (1.1, 0) – (1.9,0) ;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (3.1, 0.07) – (3.9,0.07) ;
[thick] (3.1, -0.07) – (3.9,-0.07) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (3.4,0) – (3.6,0.2);
[thick] (3.4,0) – (3.6,-0.2);

24(c-a) = 6 N + 3 ,     4(2a-c) = 12N^2-6N-3 ,

Δ= {(6k,6k+2)_k=1,…,N-1,2N} ,     𝔤 =  (𝔰𝔲_2)_4N⊕(𝔰𝔲_2)_12N-8⊕(𝔰𝔲_2)_16  .

§.§ $\ell = 4$ quotients

§.§.§ $\mathcal T^{(N)}_{\mathcal H_2,4} (2,0) $

[fill= white] (0,0) circle [radius=0.1] node[below] 1;
[fill= white] (1,0) circle [radius=0.1] node[below] 2;
[fill= white] (2,0) circle [radius=0.1] node[below] N;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N;
[thick] (0.1, 0) – (0.9,0) ;
[thick] (1.1, 0) – (1.9,0) ;
[thick] (2.1, 0.06) – (2.9,0.06) ;
[thick] (2.1, 0.02) – (2.9,0.02) ;
[thick] (2.1,-0.02) – (2.9,-0.02) ;
[thick] (2.1, -0.06) – (2.9,-0.06) ;
[thick] (2.4,0) – (2.6,0.2);
[thick] (2.4,0) – (2.6,-0.2);

24(c-a) = 3 N + 2 ,     4(2a-c) = 12N^2-19N+6 ,

Δ= {(6k,6k+2)_k=1,…,N-2,6N-6,3N/2} ,     𝔤 =  (𝔰𝔲_2)_3N⊕(𝔰𝔲_2)_16 .

§.§.§ $\mathcal T^{(N)}_{\mathcal H_2,4} (0,2) $

[fill= white] (0,0) circle [radius=0.1] node[below] 1;
[fill= white] (1,0) circle [radius=0.1] node[below] 2;
[fill= white] (2,0) circle [radius=0.1] node[below] N+1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N;
[thick] (0.1, 0) – (0.9,0) ;
[thick] (1.1, 0) – (1.9,0) ;
[thick] (2.1, 0.06) – (2.9,0.06) ;
[thick] (2.1, 0.02) – (2.9,0.02) ;
[thick] (2.1,-0.02) – (2.9,-0.02) ;
[thick] (2.1, -0.06) – (2.9,-0.06) ;
[thick] (2.4,0) – (2.6,0.2);
[thick] (2.4,0) – (2.6,-0.2);

24(c-a) = 3 N + 3 ,     4(2a-c) = 12N^2-7N-3 ,

Δ= {(6k,6k+2)_k=1,…,N-1,3N/2} ,     𝔤 =  (𝔰𝔲_2)_12N-8⊕(𝔰𝔲_2)_16 .

§.§.§ $\mathcal S^{(N)}_{\mathcal H_2,4} (1,1) $

[fill= white] (0,0) circle [radius=0.1] node[below] 1;
[fill= white] (1,0) circle [radius=0.1] node[below] 2;
[fill= white] (2,0) circle [radius=0.1] node[below] N+1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N+1;
[thick] (0.1, 0) – (0.9,0) ;
[thick] (1.1, 0) – (1.9,0) ;
[thick] (2.1, 0.06) – (2.9,0.06) ;
[thick] (2.1, 0.02) – (2.9,0.02) ;
[thick] (2.1,-0.02) – (2.9,-0.02) ;
[thick] (2.1, -0.06) – (2.9,-0.06) ;
[thick] (2.4,0) – (2.6,0.2);
[thick] (2.4,0) – (2.6,-0.2);

24(c-a) = 3 N + 4 ,     4(2a-c) = 12N^2+2N-3 ,

Δ= {(6k,6k+2)_k=1,…,N-1,6N} ,     𝔤 = (𝔰𝔲_2)_16 .

§ LIST OF 4D SCFTS FROM TYPE IIB ORBI-S-FOLDS

In this appendix we list the theories we find after the deformation of the $T(SU(k))$ tail that breaks the $\mathfrak{su}_k$ symmetry. We will call these theories with the same notation as before, adding a small circle on top to distinguish them from their progenitors.

§.§ $\ell =2$ quotients

§.§.§ $\mathring{\mathcal T}^{(N)}_{E_6,2}(2,0,0,0,0)$

[fill= white] (2,0) circle [radius=0.1] node[below] N-2;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-2;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N-2;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N-2;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N;
[fill= white] (7,0) circle [radius=0.1] node[below] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (6.1, 0.05) – (6.9,0.05) ;
[thick] (6.1, -0.05) – (6.9,-0.05) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (6.6,0) – (6.4,0.2);
[thick] (6.6,0) – (6.4,-0.2);

24(c-a) = 12N-8 ,     4(2a-c) = 12N^2-24N+16 ,

Δ= {(6k-2,6k)_k=1,…,N-2,6N-8,3N-1 } ,      𝔤 = (𝔣_4)_6N-2 .

§.§.§ $\mathring{\mathcal T}^{(N)}_{E_6,2}(1,0,0,0,1)$

[fill= white] (2,0) circle [radius=0.1] node[below] N-1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-1;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N-1;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N-1;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N;
[fill= white] (7,0) circle [radius=0.1] node[below] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (6.1, 0.05) – (6.9,0.05) ;
[thick] (6.1, -0.05) – (6.9,-0.05) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (6.6,0) – (6.4,0.2);
[thick] (6.6,0) – (6.4,-0.2);

24(c-a) = 12N-4 ,     4(2a-c) = 12N^2 -6N +1  ,

Δ= {(6k-2,6k)_k=1,…,N-1,6N-2 } ,    𝔤 = (𝔰𝔭_3)_6N-1 .

§.§.§ $\mathring{\mathcal T}^{(N)}_{E_6,2}(0,1,0,0,0)$

[fill= white] (2,0) circle [radius=0.1] node[below] N-1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-2;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N-2;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N-2;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N;
[fill= white] (7,0) circle [radius=0.1] node[below] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (6.1, 0.05) – (6.9,0.05) ;
[thick] (6.1, -0.05) – (6.9,-0.05) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (6.6,0) – (6.4,0.2);
[thick] (6.6,0) – (6.4,-0.2);

24(c-a) = 12N-7 ,     4(2a-c) = 12N^2 -12N+5 ,

Δ= {(6k-2,6k)_k=1,…,N-1,3N-1 } ,      𝔤 = (𝔰𝔬_7)_6N-2⊕(𝔰𝔲_2)_12N-12 .

§.§.§ $\mathring{\mathcal T}^{(N)}_{E_6,2}(0,0,0,0,2)$

[fill= white] (2,0) circle [radius=0.1] node[below] N;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N;
[fill= white] (7,0) circle [radius=0.1] node[below] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (6.1, 0.05) – (6.9,0.05) ;
[thick] (6.1, -0.05) – (6.9,-0.05) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (6.6,0) – (6.4,0.2);
[thick] (6.6,0) – (6.4,-0.2);

24(c-a) = 12N ,     4(2a-c) = 12N^2 ,

Δ= {(6k-2,6k)_k=1,…,N-1,(6N-2,3N) }  ,    𝔤 = (𝔰𝔭_4)_6N .

§.§.§ $\mathring{\mathcal S}^{(N)}_{E_6,2}(0,0,0,1,0)$

[fill= white] (2,0) circle [radius=0.1] node[below] N;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N+1;
[fill= white] (7,0) circle [radius=0.1] node[below] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (6.1, 0.05) – (6.9,0.05) ;
[thick] (6.1, -0.05) – (6.9,-0.05) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (6.6,0) – (6.4,0.2);
[thick] (6.6,0) – (6.4,-0.2);

24(c-a) = 12N+1 ,     4(2a-c) = 12N^2+6 ,

Δ= {(6k-2,6k)_k=1,…,N } ,    𝔤 = (𝔰𝔲_2)_6N+1⊕(𝔰𝔲_4)_12N .

§.§.§ $\mathring{\mathcal T}^{(N)}_{E_6,2}(3,0,0,0,0)$

[fill= white] (2,0) circle [radius=0.1] node[below] N-3;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-3;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N-3;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N-3;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N-1;
[fill= white] (5.5,1) circle [radius=0.1] node[above] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (5+0.05,0.1) – (5.5,0.9);
[thick] (6-0.05,0.1) – (5.5,0.9);

24(c-a) = 12N-13 ,     4(2a-c) = 18N^2 -59N+60 ,

Δ= {(6k-3,6k-1,6k)_k=1,…,N-3,(6N-15,6N-13),6N-9,3N-3/2 } ,     𝔤 = (𝔣_4)_6N-3 .

§.§.§ $\mathring{\mathcal T}^{(N)}_{E_6,2}(2,0,0,0,1)$

[fill= white] (2,0) circle [radius=0.1] node[below] N-2;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-2;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N-2;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N-2;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N-1;
[fill= white] (5.5,1) circle [radius=0.1] node[above] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (5+0.05,0.1) – (5.5,0.9);
[thick] (6-0.05,0.1) – (5.5,0.9);

24(c-a) = 12N-9 ,     4(2a-c) = 18N^2-29N+17 ,

Δ= {(6k-3,6k-1,6k)_k=1,…,N-2,(6N-9,6N-7),6N-3 } ,     𝔤 = (𝔰𝔭_3)_6N-2 .

§.§.§ $\mathring{\mathcal T}^{(N)}_{E_6,2}(1,0,0,0,2)$

[fill= white] (2,0) circle [radius=0.1] node[below] N-1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-1;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N-1;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N-1;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N-1;
[fill= white] (5.5,1) circle [radius=0.1] node[above] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (5+0.05,0.1) – (5.5,0.9);
[thick] (6-0.05,0.1) – (5.5,0.9);

24(c-a) = 12N-5 ,     4(2a-c) = 18N^2-11N+2 ,

Δ= {(6k-3,6k-1,6k)_k=1,…,N-1,(6N-3,3N-1/2) } ,     𝔤 = (𝔰𝔭_3)_6N-1 .

§.§.§ $\mathring{\mathcal T}^{(N)}_{E_6,2}(1,1,0,0,0)$

[fill= white] (2,0) circle [radius=0.1] node[below] N-2;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-3;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N-3;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N-3;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N-1;
[fill= white] (5.5,1) circle [radius=0.1] node[above] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (5+0.05,0.1) – (5.5,0.9);
[thick] (6-0.05,0.1) – (5.5,0.9);

24(c-a) = 12N-12 ,     4(2a-c) = 18N^2-35N+20 ,

Δ= {(6k-3,6k-1,6k)_k=1,…,N-2,(6N-9,6N-7),3N-3/2 } ,     𝔤 = (𝔰𝔬_7)_6N-3 .

§.§.§ $\mathring{\mathcal S}^{(N)}_{E_6,2}(1,0,0,1,0)$

[fill= white] (2,0) circle [radius=0.1] node[below] N-1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-1;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N-1;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N-1;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N;
[fill= white] (5.5,1) circle [radius=0.1] node[above] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (5+0.05,0.1) – (5.5,0.9);
[thick] (6-0.05,0.1) – (5.5,0.9);

24(c-a) = 12N-4 ,     4(2a-c) = 18N^2 -5N+1 ,

Δ= {(6k-3,6k-1,6k)_k=1,…,N-1,(6N-3,6N-1) }  ,     𝔤 = (𝔰𝔲_2)_6N ⊕(𝔰𝔲_3)_12N-2 .

§.§.§ $\mathring{\mathcal T}^{(N)}_{E_6,2}(0,0,0,0,3)$

[fill= white] (2,0) circle [radius=0.1] node[below] N;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N-1;
[fill= white] (5.5,1) circle [radius=0.1] node[above] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (5+0.05,0.1) – (5.5,0.9);
[thick] (6-0.05,0.1) – (5.5,0.9);

24(c-a) = 12N-1 ,     4(2a-c) = 18N^2 -5N+1 ,

Δ= {(6k-3,6k-1,6k)_k=1,…,N-1,(6N-3,6N-1) } ,     𝔤 = (𝔰𝔭_4)_6N

§.§.§ $\mathring{\mathcal T}^{(N)}_{E_6,2}(0,1,0,0,1)$

[fill= white] (2,0) circle [radius=0.1] node[below] N-1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-2;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N-2;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N-2;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N-1;
[fill= white] (5.5,1) circle [radius=0.1] node[above] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (5+0.05,0.1) – (5.5,0.9);
[thick] (6-0.05,0.1) – (5.5,0.9);

24(c-a) = 12N-8 ,     4(2a-c) = 18N^2-17N+4 ,

Δ= {(6k-3,6k-1,6k)_k=1,…,N-1,6N-3 }    𝔤 = (𝔰𝔭_2)_6N-2 ⊕(𝔰𝔲_2)_12N-12 .

§.§.§ $\mathring{\mathcal T}^{(N)}_{E_6,2}(0,0,0,1,1)$

[fill= white] (2,0) circle [radius=0.1] node[below] N;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N;
[fill= white] (5.5,1) circle [radius=0.1] node[above] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (5+0.05,0.1) – (5.5,0.9);
[thick] (6-0.05,0.1) – (5.5,0.9);

24(c-a) = 12N ,     4(2a-c) = 18N^2+7N ,

Δ= {(6k-3,6k-1,6k)_k=1,…,N } ,     𝔤 = (𝔰𝔲_4)_12N .

§.§.§ $\mathring{\mathcal T}^{(N)}_{E_6,2}(0,0,1,0,0)$

[fill= white] (2,0) circle [radius=0.1] node[below] N-1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-2;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N-3;
[fill= white] (5,0) circle [radius=0.1] node[below] 4N-3;
[fill= white] (6,0) circle [radius=0.1] node[below] 2N-1;
[fill= white] (5.5,1) circle [radius=0.1] node[above] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05) ;
[thick] (4.1, -0.05) – (4.9,-0.05) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (5.1, 0) – (5.9,0) ;
[thick] (4.4,0) – (4.6,0.2);
[thick] (4.4,0) – (4.6,-0.2);
[thick] (5+0.05,0.1) – (5.5,0.9);
[thick] (6-0.05,0.1) – (5.5,0.9);

24(c-a) = 12N-10 ,     4(2a-c) = 18N^2 -23N+7 ,

Δ= {(6k-3,6k-1,6k)_k=1,…,N-1,3N-3/2 } ,     𝔤 = (𝔰𝔲_3)_6N-3 ⊕(𝔰𝔲_3)_12N-12

§.§ $\ell =3$ quotients

§.§.§ $\mathring{\mathcal T}^{(N)}_{D_4,3}(2,0,0)$

[fill= white] (2,0) circle [radius=0.1] node[below] N-2;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-2;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N-2;
[fill= white] (5,0) circle [radius=0.1] node[below] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (3.1, 0.07) – (3.9,0.07) ;
[thick] (3.1, -0.07) – (3.9,-0.07) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05);
[thick] (4.1, -0.05) – (4.9,-0.05);
[thick] (3.4,0) – (3.6,0.2);
[thick] (3.4,0) – (3.6,-0.2);

24(c-a) = 12N-6 ,     4(2a-c) = 12N^2 -26N + 50/3 ,

Δ= {(6k-2,6k)_k=1,…,N-2,6N-8,2N-2/3 } ,     𝔤 =  (𝔤_2)_4N-4/3 .

§.§.§ $\mathring{\mathcal S}^{(N)}_{D_4,3}(1,0,1)$

[fill= white] (2,0) circle [radius=0.1] node[below] N-1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-1;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N-1;
[fill= white] (5,0) circle [radius=0.1] node[below] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (3.1, 0.07) – (3.9,0.07) ;
[thick] (3.1, -0.07) – (3.9,-0.07) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05);
[thick] (4.1, -0.05) – (4.9,-0.05);
[thick] (3.4,0) – (3.6,0.2);
[thick] (3.4,0) – (3.6,-0.2);

24(c-a) = 12N-6 ,     4(2a-c) = 12N^2-6N+1 ,

Δ= {(6k-2,6k)_k=1,…,N-1,6N-2} ,     𝔤 =  (𝔰𝔲_2)_12N-2 .

§.§.§ $\mathring{\mathcal R}^{(N)}_{D_4,3}(0,0,2)$

[fill= white] (2,0) circle [radius=0.1] node[below] N;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N;
[fill= white] (5,0) circle [radius=0.1] node[below] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (3.1, 0.07) – (3.9,0.07) ;
[thick] (3.1, -0.07) – (3.9,-0.07) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05);
[thick] (4.1, -0.05) – (4.9,-0.05);
[thick] (3.4,0) – (3.6,0.2);
[thick] (3.4,0) – (3.6,-0.2);

24(c-a) = 12N ,     4(2a-c) = 12N^2+6N ,

Δ= {(6k-2,6k)_k=1,…,N} ,     𝔤 =  (𝔰𝔲_3)_12N .

§.§.§ $\mathring{\mathcal T}^{(N)}_{D_4,3}(0,1,0)$

[fill= white] (2,0) circle [radius=0.1] node[below] N-1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-2;
[fill= white] (4,0) circle [radius=0.1] node[below] 3N-2;
[fill= white] (5,0) circle [radius=0.1] node[below] 1;
[thick] (2.1, 0) – (2.9,0) ;
[thick] (3.1, 0.07) – (3.9,0.07) ;
[thick] (3.1, -0.07) – (3.9,-0.07) ;
[thick] (3.1, 0) – (3.9,0) ;
[thick] (4.1, 0.05) – (4.9,0.05);
[thick] (4.1, -0.05) – (4.9,-0.05);
[thick] (3.4,0) – (3.6,0.2);
[thick] (3.4,0) – (3.6,-0.2);

24(c-a) = 12N-5 ,     4(2a-c) = 12N^2-14 N +11/3 ,

Δ= {(6k-2,6k)_k=1,…,N-1,2N-2/3} ,     𝔤 =  (𝔰𝔲_2)_4N-2/3⊕(𝔰𝔲_2)_12N-12 .

§.§ $\ell = 4$ quotients

§.§.§ $\mathring{\mathcal T}^{(N)}_{\mathcal H_2,4} (2,0) $

[fill= white] (2,0) circle [radius=0.1] node[below] N-2;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-2;
[fill= white] (4,0) circle [radius=0.1] node[below] 1;
[thick] (2.1, 0.06) – (2.9,0.06) ;
[thick] (2.1, 0.02) – (2.9,0.02) ;
[thick] (2.1,-0.02) – (2.9,-0.02) ;
[thick] (2.1, -0.06) – (2.9,-0.06) ;
[thick] (3.1, 0.06) – (3.9,0.06) ;
[thick] (3.1, 0.02) – (3.9,0.02) ;
[thick](3.1,-0.02) – (3.9,-0.02) ;
[thick] (3.1, -0.06) – (3.9,-0.06) ;
[thick] (2.4,0) – (2.6,0.2);
[thick] (2.4,0) – (2.6,-0.2);

24(c-a) = 3N-5 ,     4(2a-c) = 12N^2 -27N +17 ,

Δ= {(6k-2,6k)_k=1,…,N-2,6N-8,3/2 N -1/2 } ,     𝔤 =  (𝔰𝔲_2)_3N-3 .

§.§.§ $\mathring{\mathcal T}^{(N)}_{\mathcal H_2,4} (0,2) $

[fill= white] (2,0) circle [radius=0.1] node[below] N-1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-2;
[fill= white] (4,0) circle [radius=0.1] node[below] 1;
[thick] (2.1, 0.06) – (2.9,0.06) ;
[thick] (2.1, 0.02) – (2.9,0.02) ;
[thick] (2.1,-0.02) – (2.9,-0.02) ;
[thick] (2.1, -0.06) – (2.9,-0.06) ;
[thick] (3.1, 0.06) – (3.9,0.06) ;
[thick](3.1, 0.02) – (3.9,0.02) ;
[thick] (3.1,-0.02) – (3.9,-0.02) ;
[thick] (3.1, -0.06) – (3.9,-0.06) ;
[thick] (2.4,0) – (2.6,0.2);
[thick] (2.4,0) – (2.6,-0.2);

24(c-a) = 3N-4 ,     4(2a-c) = 12N^2 -15 N +4  ,

Δ= {(6k-2,6k)_k=1,…,N-1,3/2N-1/2 } ,     𝔤 =  (𝔰𝔲_2)_12N-12 .

§.§.§ $\mathring{\mathcal S}^{(N)}_{\mathcal H_2,4} (1,1) $

[fill= white] (2,0) circle [radius=0.1] node[below] N-1;
[fill= white] (3,0) circle [radius=0.1] node[below] 2N-1;
[fill= white] (4,0) circle [radius=0.1] node[below] 1;
[thick] (2.1, 0.06) – (2.9,0.06) ;
[thick] (2.1, 0.02) – (2.9,0.02) ;
[thick] (2.1,-0.02) – (2.9,-0.02) ;
[thick] (2.1, -0.06) – (2.9,-0.06) ;
[thick] (3.1, 0.06) – (3.9,0.06) ;
[thick](3.1, 0.02) – (3.9,0.02) ;
[thick] (3.1,-0.02) – (3.9,-0.02) ;
[thick] (3.1, -0.06) – (3.9,-0.06) ;
[thick] (2.4,0) – (2.6,0.2);
[thick] (2.4,0) – (2.6,-0.2);

24(c-a) = 3N-3 ,     4(2a-c) = 12N^2 -6N+1 ,

Δ= {(6k-2,6k)_k=1,…,N-1,6N-2 } ,     𝔤 = ∅ .