$12^{2}_{252}$ - Minimal pinning sets
Pinning sets for 12^2_252 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_252 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 168
of which optimal: 4
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03427
on average over minimal pinning sets: 2.51429
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
B (optimal)
• {1, 2, 6, 7, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
C (optimal)
• {1, 2, 6, 7, 9, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
D (optimal)
• {1, 2, 5, 6, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
a (minimal)
• {1, 2, 4, 5, 7, 9, 12}
7
[2, 2, 2, 3, 3, 3, 3]
2.57
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
4
0
0
2.5
7
0
1
21
2.75
8
0
0
48
2.95
9
0
0
53
3.11
10
0
0
31
3.22
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
1
163
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,2],[0,1,5,6],[0,6,6,0],[1,7,8,5],[1,4,8,2],[2,9,3,3],[4,9,9,8],[4,7,9,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,16,10,17],[17,7,18,8],[19,1,20,2],[5,15,6,16],[10,6,11,7],[18,3,19,2],[4,12,5,13],[14,11,15,12],[3,14,4,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(11,2,-12,-3)(13,4,-14,-5)(5,12,-6,-13)(7,18,-8,-19)(19,6,-20,-7)(8,9,-1,-10)(15,10,-16,-11)(3,14,-4,-15)(20,17,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,10)(-2,11,-16)(-3,-15,-11)(-4,13,-6,19,-8,-10,15)(-5,-13)(-7,-19)(-9,8,18)(-12,5,-14,3)(-17,20,6,12,2)(-18,7,-20)(1,9,17)(4,14)
Multiloop annotated with half-edges
12^2_252 annotated with half-edges