$12^{2}_{80}$ - Minimal pinning sets
Pinning sets for 12^2_80 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_80 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],[0,3,4,0],[0,4,4,5],[0,5,6,1],[1,7,2,2],[2,8,6,3],[3,5,9,9],[4,9,8,8],[5,7,7,9],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,15,18,16],[19,9,20,10],[1,14,2,15],[18,11,19,10],[13,8,14,9],[2,8,3,7],[11,4,12,5],[5,12,6,13],[3,6,4,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,2,-10,-3)(1,4,-2,-5)(5,16,-6,-1)(11,6,-12,-7)(18,7,-19,-8)(8,17,-9,-18)(3,10,-4,-11)(15,12,-16,-13)(20,13,-17,-14)(14,19,-15,-20)
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,-5)(-2,9,17,13,-16,5)(-3,-11,-7,18,-9)(-4,1,-6,11)(-8,-18)(-10,3)(-12,15,19,7)(-13,20,-15)(-14,-20)(-17,8,-19,14)(2,4,10)(6,16,12)
Multiloop annotated with half-edges
12^2_80 annotated with half-edges