$12^{2}_{188}$ - Minimal pinning sets
Pinning sets for 12^2_188 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_188 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96564
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 9}
4
[2, 2, 2, 2]
2.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,5,6,0],[0,7,8,4],[1,3,5,1],[1,4,9,2],[2,9,7,7],[3,6,6,8],[3,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[16,9,1,10],[10,6,11,5],[8,15,9,16],[1,13,2,12],[6,12,7,11],[7,4,8,5],[14,20,15,17],[13,20,14,19],[2,19,3,18],[3,17,4,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,2,-6,-3)(3,8,-4,-9)(14,7,-15,-8)(9,4,-10,-5)(10,15,-11,-16)(6,13,-7,-14)(17,16,-18,-1)(18,11,-19,-12)(12,19,-13,-20)(1,20,-2,-17)
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,-17)(-2,5,-10,-16,17)(-3,-9,-5)(-4,9)(-6,-14,-8,3)(-7,14)(-11,18,16)(-12,-20,1,-18)(-13,6,2,20)(-15,10,4,8)(-19,12)(7,13,19,11,15)
Multiloop annotated with half-edges
12^2_188 annotated with half-edges