$12^{1}_{266}$ - Minimal pinning sets
Pinning sets for 12^1_266 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_266 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 156
of which optimal: 4
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98542
on average over minimal pinning sets: 2.46429
on average over optimal pinning sets: 2.375
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 7, 10, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 3, 4, 7, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
C (optimal)
• {1, 3, 4, 6, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
D (optimal)
• {1, 2, 4, 6, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
a (minimal)
• {1, 2, 4, 5, 7, 10, 11}
7
[2, 2, 2, 2, 3, 3, 4]
2.57
b (minimal)
• {1, 2, 4, 5, 7, 9, 11}
7
[2, 2, 2, 2, 3, 4, 4]
2.71
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
4
0
0
2.38
7
0
2
20
2.68
8
0
0
46
2.91
9
0
0
48
3.08
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
2
150
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,5],[0,6,6,3],[0,2,7,0],[1,7,8,5],[1,4,8,1],[2,9,7,2],[3,6,9,4],[4,9,9,5],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,13,1,14],[14,8,15,7],[19,2,20,3],[12,1,13,2],[8,5,9,6],[15,6,16,7],[3,18,4,19],[4,11,5,12],[9,17,10,16],[10,17,11,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,2,-6,-3)(12,3,-13,-4)(4,11,-5,-12)(1,8,-2,-9)(17,10,-18,-11)(13,6,-14,-7)(7,14,-8,-15)(20,15,-1,-16)(9,18,-10,-19)(16,19,-17,-20)
Edge permutation $\epsilon=$ (-1,1)(-2,2)(-3,3)(-4,4)(-5,5)(-6,6)(-7,7)(-8,8)(-9,9)(-10,10)(-11,11)(-12,12)(-13,13)(-14,14)(-15,15)(-16,16)(-17,17)(-18,18)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-9,-19,16)(-2,5,11,-18,9)(-3,12,-5)(-4,-12)(-6,13,3)(-7,-15,20,-17,-11,4,-13)(-8,1,15)(-10,17,19)(-14,7)(-16,-20)(2,8,14,6)(10,18)
Loop annotated with half-edges
12^1_266 annotated with half-edges