$12^{2}_{173}$ - Minimal pinning sets
Pinning sets for 12^2_173 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_173 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 352
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04132
on average over minimal pinning sets: 2.41667
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 8, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 6, 9, 10}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 2, 4, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
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
2.25
5
0
2
8
2.56
6
0
0
39
2.79
7
0
0
81
2.95
8
0
0
100
3.08
9
0
0
76
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
2
349
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 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],[0,6,7,8],[0,8,6,5],[1,4,2,1],[2,4,7,3],[3,6,9,9],[3,9,9,4],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,10,14,9],[11,8,12,9],[3,19,4,20],[1,16,2,15],[10,15,11,14],[2,7,3,8],[18,6,19,7],[4,17,5,16],[5,17,6,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,4,-16,-5)(12,5,-1,-6)(9,6,-10,-7)(7,18,-8,-19)(19,8,-20,-9)(20,11,-13,-12)(13,2,-14,-3)(3,14,-4,-15)(1,16,-2,-17)(10,17,-11,-18)
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,-17,10,6)(-2,13,11,17)(-3,-15,-5,12,-13)(-4,15)(-6,9,-20,-12)(-7,-19,-9)(-8,19)(-10,-18,7)(-11,20,8,18)(-14,3)(-16,1,5)(2,16,4,14)
Multiloop annotated with half-edges
12^2_173 annotated with half-edges