$12^{2}_{219}$ - Minimal pinning sets
Pinning sets for 12^2_219 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_219 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 120
of which optimal: 2
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9731
on average over minimal pinning sets: 2.38095
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)
• {1, 2, 3, 5, 7, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 4, 7, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
a (minimal)
• {1, 2, 3, 5, 7, 8, 11}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
b (minimal)
• {1, 2, 3, 4, 7, 8, 11}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.33
7
0
2
11
2.6
8
0
0
32
2.84
9
0
0
39
3.03
10
0
0
25
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
2
116
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,3],[0,2,6,6],[0,7,8,8],[1,9,9,1],[2,7,3,3],[4,6,9,8],[4,7,9,4],[5,8,7,5]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,5,12,6],[6,9,7,10],[7,19,8,20],[1,15,2,16],[4,12,5,13],[18,8,19,9],[14,17,15,18],[2,17,3,16],[13,3,14,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-11,-2)(17,4,-18,-5)(2,5,-3,-6)(12,9,-13,-10)(10,11,-1,-12)(7,14,-8,-15)(15,8,-16,-9)(13,16,-14,-17)(3,18,-4,-19)(6,19,-7,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,-7,-15,-9,12)(-2,-6,-20)(-3,-19,6)(-4,17,-14,7,19)(-5,2,-11,10,-13,-17)(-8,15)(-10,-12)(-16,13,9)(-18,3,5)(1,11)(4,18)(8,14,16)
Multiloop annotated with half-edges
12^2_219 annotated with half-edges