$12^{2}_{172}$ - Minimal pinning sets
Pinning sets for 12^2_172 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_172 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
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: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 4, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,7],[0,8,9,5],[1,4,2,1],[2,9,9,3],[3,8,8,3],[4,7,7,9],[4,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,6,10,5],[7,4,8,5],[19,14,20,15],[1,12,2,11],[6,11,7,10],[3,15,4,16],[13,18,14,19],[12,18,13,17],[2,17,3,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,2,-6,-3)(3,14,-4,-15)(15,4,-16,-5)(16,7,-17,-8)(12,19,-13,-20)(1,20,-2,-9)(9,8,-10,-1)(10,17,-11,-18)(18,11,-19,-12)(6,13,-7,-14)
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)(-2,5,-16,-8,9)(-3,-15,-5)(-4,15)(-6,-14,3)(-7,16,4,14)(-10,-18,-12,-20,1)(-11,18)(-13,6,2,20)(-17,10,8)(-19,12)(7,13,19,11,17)
Multiloop annotated with half-edges
12^2_172 annotated with half-edges