$12^{2}_{171}$ - Minimal pinning sets
Pinning sets for 12^2_171 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_171 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 176
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97631
on average over minimal pinning sets: 2.34444
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, 6, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 5, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 4, 5, 7, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
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
2.2
6
0
2
7
2.5
7
0
0
30
2.75
8
0
0
51
2.94
9
0
0
49
3.08
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
2
173
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: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,7],[0,8,6,5],[1,4,2,1],[2,4,8,3],[3,9,9,3],[4,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,10,14,9],[11,8,12,9],[3,19,4,20],[1,16,2,15],[10,15,11,14],[2,7,3,8],[18,4,19,5],[16,6,17,7],[5,17,6,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,12,-14,-1)(19,2,-20,-3)(10,3,-11,-4)(7,4,-8,-5)(5,16,-6,-17)(1,20,-2,-13)(11,14,-12,-15)(8,15,-9,-16)(17,6,-18,-7)(18,9,-19,-10)
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,-13)(-2,19,9,15,-12,13)(-3,10,-19)(-4,7,-18,-10)(-5,-17,-7)(-6,17)(-8,-16,5)(-9,18,6,16)(-11,-15,8,4)(-14,11,3,-20,1)(2,20)(12,14)
Multiloop annotated with half-edges
12^2_171 annotated with half-edges