$12^{2}_{177}$ - Minimal pinning sets
Pinning sets for 12^2_177 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_177 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 288
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04277
on average over minimal pinning sets: 2.45
on average over optimal pinning sets: 2.45
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 5, 7, 9}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 3, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.45
6
0
0
24
2.72
7
0
0
61
2.91
8
0
0
85
3.05
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
0
284
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,5,6,0],[0,7,8,4],[1,3,5,1],[1,4,8,2],[2,9,9,7],[3,6,9,8],[3,7,9,5],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,17,8,16],[19,5,20,6],[1,10,2,9],[17,9,18,8],[18,15,19,16],[12,4,13,5],[10,13,11,14],[2,14,3,15],[3,11,4,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(15,2,-16,-3)(20,9,-7,-10)(5,10,-6,-11)(14,11,-15,-12)(12,17,-13,-18)(3,16,-4,-17)(18,13,-19,-14)(19,4,-20,-5)(6,7,-1,-8)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,15,11,-6,-8)(-3,-17,12,-15)(-4,19,13,17)(-5,-11,14,-19)(-7,6,10)(-9,20,4,16,2)(-10,5,-20)(-12,-18,-14)(-13,18)(-16,3)(1,7,9)
Multiloop annotated with half-edges
12^2_177 annotated with half-edges