$12^{2}_{158}$ - Minimal pinning sets
Pinning sets for 12^2_158 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_158 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 224
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9785
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 4, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
C (optimal)
• {1, 3, 5, 7, 11}
5
[2, 2, 2, 2, 4]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.27
6
0
0
18
2.59
7
0
0
46
2.82
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
221
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: [[1,2,3,4],[0,5,5,6],[0,7,7,8],[0,8,4,4],[0,3,3,5],[1,4,9,1],[1,9,9,7],[2,6,8,2],[2,7,9,3],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[16,3,1,4],[4,17,5,20],[10,15,11,16],[2,7,3,8],[1,7,2,6],[17,6,18,5],[13,19,14,20],[14,9,15,10],[11,9,12,8],[18,12,19,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(3,18,-4,-19)(4,15,-5,-16)(16,5,-1,-6)(6,9,-7,-10)(12,7,-13,-8)(19,10,-20,-11)(8,13,-9,-14)(11,20,-12,-17)(17,2,-18,-3)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-9,6)(-2,17,-12,-8,-14)(-3,-19,-11,-17)(-4,-16,-6,-10,19)(-5,16)(-7,12,20,10)(-13,8)(-15,4,18,2)(-18,3)(-20,11)(1,5,15)(7,9,13)
Multiloop annotated with half-edges
12^2_158 annotated with half-edges