$12^{2}_{227}$ - Minimal pinning sets
Pinning sets for 12^2_227 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_227 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 7, 11}
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
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,5],[0,6,7,4],[1,3,7,8],[2,9,6,2],[3,5,9,9],[3,8,8,4],[4,7,7,9],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,11,14,12],[14,19,15,20],[1,6,2,7],[7,10,8,11],[18,15,19,16],[5,17,6,18],[2,9,3,10],[8,3,9,4],[16,4,17,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,12,-14,-1)(9,2,-10,-3)(10,5,-11,-6)(3,6,-4,-7)(18,7,-19,-8)(4,11,-5,-12)(15,20,-16,-13)(1,14,-2,-15)(19,16,-20,-17)(8,17,-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,-15,-13)(-2,9,17,-20,15)(-3,-7,18,-9)(-4,-12,13,-16,19,7)(-5,10,2,14,12)(-6,3,-10)(-8,-18)(-11,4,6)(-14,1)(-17,8,-19)(5,11)(16,20)
Multiloop annotated with half-edges
12^2_227 annotated with half-edges