$12^{2}_{231}$ - Minimal pinning sets
Pinning sets for 12^2_231 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_231 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 7, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 5, 8, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 5, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,6,6,7],[0,8,9,0],[1,9,5,1],[1,4,6,6],[2,5,5,2],[2,9,8,8],[3,7,7,9],[3,8,7,4]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,8,14,7],[11,4,12,5],[19,1,20,2],[8,15,9,14],[9,6,10,7],[5,10,6,11],[3,17,4,18],[2,17,3,16],[18,15,19,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,12,-8,-1)(5,2,-6,-3)(3,20,-4,-13)(1,6,-2,-7)(18,9,-19,-10)(13,4,-14,-5)(17,14,-18,-15)(15,10,-16,-11)(11,16,-12,-17)(8,19,-9,-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,-7)(-2,5,-14,17,-12,7)(-3,-13,-5)(-4,13)(-6,1,-8,-20,3)(-9,18,14,4,20)(-10,15,-18)(-11,-17,-15)(-16,11)(-19,8,12,16,10)(2,6)(9,19)
Multiloop annotated with half-edges
12^2_231 annotated with half-edges