$12^{2}_{218}$ - Minimal pinning sets
Pinning sets for 12^2_218 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_218 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 200
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03442
on average over minimal pinning sets: 2.49286
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 5, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 3, 5, 7, 9, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 2, 4, 5, 7, 9, 12}
7
[2, 2, 2, 3, 3, 3, 3]
2.57
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.4
6
0
2
7
2.63
7
0
1
31
2.82
8
0
0
58
2.99
9
0
0
58
3.12
10
0
0
32
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
3
196
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 5, 5, 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],[0,7,7,8],[1,8,9,1],[2,9,3,3],[4,9,8,4],[4,7,9,5],[5,8,7,6]]
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],[14,2,15,3],[16,14,17,13],[17,3,18,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(15,2,-16,-3)(12,5,-13,-6)(3,6,-4,-7)(18,9,-19,-10)(1,16,-2,-17)(17,20,-18,-11)(8,19,-9,-20)(4,13,-5,-14)(7,14,-8,-15)
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,-17,-11)(-2,15,-8,-20,17)(-3,-7,-15)(-4,-14,7)(-5,12,10,-19,8,14)(-6,3,-16,1,-12)(-9,18,20)(-10,11,-18)(-13,4,6)(2,16)(5,13)(9,19)
Multiloop annotated with half-edges
12^2_218 annotated with half-edges