$12^{2}_{148}$ - Minimal pinning sets
Pinning sets for 12^2_148 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_148 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 252
of which optimal: 6
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97224
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 10, 11}
5
[2, 2, 2, 2, 4]
2.40
B (optimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
C (optimal)
• {1, 3, 4, 8, 11}
5
[2, 2, 2, 2, 4]
2.40
D (optimal)
• {1, 3, 4, 5, 11}
5
[2, 2, 2, 2, 4]
2.40
E (optimal)
• {1, 2, 3, 4, 11}
5
[2, 2, 2, 2, 3]
2.20
F (optimal)
• {1, 3, 4, 9, 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
6
0
0
2.33
6
0
0
27
2.65
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
6
0
246
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,6,6,7],[0,7,1,0],[1,7,8,5],[1,4,9,9],[2,8,7,2],[2,6,4,3],[4,6,9,9],[5,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,19,14,18],[4,11,5,12],[19,1,20,2],[14,6,15,7],[7,17,8,18],[10,3,11,4],[5,3,6,2],[15,10,16,9],[16,8,17,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,12,-8,-1)(18,3,-19,-4)(20,5,-13,-6)(6,19,-7,-20)(1,8,-2,-9)(15,10,-16,-11)(11,2,-12,-3)(4,13,-5,-14)(14,17,-15,-18)(9,16,-10,-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,-9,-17,14,-5,20,-7)(-2,11,-16,9)(-3,18,-15,-11)(-4,-14,-18)(-6,-20)(-8,1)(-10,15,17)(-12,7,19,3)(-13,4,-19,6)(2,8,12)(5,13)(10,16)
Multiloop annotated with half-edges
12^2_148 annotated with half-edges