$12^{2}_{91}$ - Minimal pinning sets
Pinning sets for 12^2_91 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_91 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 448
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: 3.04602
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)
• {2, 4, 8, 9}
4
[2, 2, 2, 4]
2.50
B (optimal)
• {2, 4, 7, 9}
4
[2, 2, 2, 3]
2.25
C (optimal)
• {2, 4, 9, 10}
4
[2, 2, 2, 3]
2.25
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
3
0
0
2.33
5
0
0
21
2.65
6
0
0
64
2.85
7
0
0
111
3.0
8
0
0
120
3.1
9
0
0
83
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
0
445
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,7,7],[0,7,8,9],[0,5,1,1],[1,4,9,6],[2,5,8,7],[2,6,3,2],[3,6,9,9],[3,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,20,3,11],[9,15,10,16],[4,8,5,7],[1,12,2,11],[12,19,13,20],[16,13,17,14],[14,8,15,9],[5,17,6,18],[18,6,19,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (19,4,-20,-5)(5,2,-6,-3)(15,6,-16,-7)(1,8,-2,-9)(17,14,-18,-15)(7,16,-8,-17)(3,18,-4,-19)(13,20,-14,-11)(10,11,-1,-12)(12,9,-13,-10)
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,-9,12)(-2,5,-20,13,9)(-3,-19,-5)(-4,19)(-6,15,-18,3)(-7,-17,-15)(-8,1,11,-14,17)(-10,-12)(-11,10,-13)(-16,7)(2,8,16,6)(4,18,14,20)
Multiloop annotated with half-edges
12^2_91 annotated with half-edges