$12^{2}_{222}$ - Minimal pinning sets
Pinning sets for 12^2_222 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_222 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, 2, 4, 7, 11}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 3, 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
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, 3, 4, 4, 4, 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],[0,5,6,4],[1,3,6,7],[2,8,3,2],[3,9,9,4],[4,9,8,8],[5,7,7,9],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,9,12,10],[12,19,13,20],[1,18,2,17],[8,16,9,17],[18,13,19,14],[2,7,3,8],[15,5,16,6],[14,5,15,4],[6,3,7,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (4,1,-5,-2)(2,11,-3,-12)(12,3,-13,-4)(10,5,-1,-6)(18,7,-19,-8)(15,20,-16,-11)(6,13,-7,-14)(14,9,-15,-10)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,-13,6)(-2,-12,-4)(-3,12)(-5,10,-15,-11,2)(-6,-14,-10)(-7,18,-9,14)(-8,-18)(-16,19,7,13,3,11)(-17,8,-19)(-20,15,9,17)(1,5)(16,20)
Multiloop annotated with half-edges
12^2_222 annotated with half-edges