$12^{2}_{61}$ - Minimal pinning sets
Pinning sets for 12^2_61 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_61 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, 2, 4, 6, 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)
• {2, 3, 4, 6, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {2, 3, 4, 5, 8, 9, 11}
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
3.22
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, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,7,3],[0,2,4,4],[0,3,3,1],[1,7,8,1],[2,8,9,9],[2,9,8,5],[5,7,9,6],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[12,3,1,4],[4,13,5,20],[11,15,12,16],[2,14,3,15],[1,14,2,13],[5,19,6,20],[16,9,17,8],[18,10,19,11],[6,10,7,9],[17,7,18,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(16,5,-17,-6)(6,15,-7,-16)(7,4,-8,-5)(17,8,-18,-9)(3,18,-4,-19)(19,2,-20,-3)(20,11,-13,-12)(12,13,-1,-14)(9,14,-10,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,14)(-2,19,-4,7,15,-10)(-3,-19)(-5,16,-7)(-6,-16)(-8,17,5)(-9,-15,6,-17)(-11,20,2)(-12,-14,9,-18,3,-20)(-13,12)(1,13,11)(4,18,8)
Multiloop annotated with half-edges
12^2_61 annotated with half-edges