$12^{2}_{145}$ - Minimal pinning sets
Pinning sets for 12^2_145 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_145 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 240
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97778
on average over minimal pinning sets: 2.3
on average over optimal pinning sets: 2.3
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 6, 9, 12}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 3, 6, 8, 9}
5
[2, 2, 2, 2, 4]
2.40
D (optimal)
• {1, 2, 3, 6, 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
4
0
0
2.3
6
0
0
22
2.62
7
0
0
52
2.84
8
0
0
69
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
4
0
236
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,6,7,8],[0,8,1,0],[1,9,9,5],[1,4,6,6],[2,5,5,9],[2,9,8,8],[2,7,7,3],[4,7,6,4]]
PD code (use to draw this multiloop with SnapPy): [[7,16,8,1],[9,6,10,7],[12,15,13,16],[8,2,9,1],[5,20,6,17],[10,20,11,19],[11,18,12,19],[14,3,15,4],[13,3,14,2],[17,4,18,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,11,-1,-12)(12,1,-13,-2)(17,2,-18,-3)(7,4,-8,-5)(10,13,-11,-14)(5,14,-6,-15)(15,6,-16,-7)(19,8,-20,-9)(3,20,-4,-17)(9,18,-10,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,17,-4,7,-16,-12)(-3,-17)(-5,-15,-7)(-6,15)(-8,19,-10,-14,5)(-9,-19)(-11,16,6,14)(-13,10,18,2)(-18,9,-20,3)(1,11,13)(4,20,8)
Multiloop annotated with half-edges
12^2_145 annotated with half-edges