$12^{2}_{81}$ - Minimal pinning sets
Pinning sets for 12^2_81 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_81 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)
• {1, 2, 5, 7, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 7, 8}
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, 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],[0,6,4,4],[1,3,3,7],[2,8,6,2],[3,5,9,7],[4,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,3,6,4],[6,19,7,20],[1,11,2,12],[12,2,13,3],[18,7,19,8],[10,17,11,18],[13,17,14,16],[8,16,9,15],[9,14,10,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(14,1,-15,-2)(11,16,-12,-17)(3,20,-4,-5)(18,7,-19,-8)(8,17,-9,-18)(9,6,-10,-7)(19,10,-20,-11)(15,12,-16,-13)(2,13,-3,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-3,-5)(-2,-14)(-4,5)(-6,9,17,-12,15,1)(-7,18,-9)(-8,-18)(-10,19,7)(-11,-17,8,-19)(-13,2,-15)(-16,11,-20,3,13)(4,20,10,6)(12,16)
Multiloop annotated with half-edges
12^2_81 annotated with half-edges