$12^{2}_{174}$ - Minimal pinning sets
Pinning sets for 12^2_174 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_174 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 272
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0386
on average over minimal pinning sets: 2.46667
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, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 4, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 5, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 5, 6, 9, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.4
6
0
1
19
2.68
7
0
0
55
2.89
8
0
0
81
3.04
9
0
0
69
3.15
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
1
268
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: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,8],[0,9,9,5],[1,4,2,1],[2,9,7,3],[3,6,8,8],[3,7,7,9],[4,8,6,4]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,8,12,7],[9,6,10,7],[16,19,17,20],[1,14,2,13],[8,13,9,12],[15,5,16,6],[18,4,19,5],[17,4,18,3],[14,3,15,2]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,11,-1,-12)(1,14,-2,-15)(15,2,-16,-3)(12,3,-13,-4)(7,4,-8,-5)(5,18,-6,-19)(19,6,-20,-7)(20,9,-11,-10)(13,16,-14,-17)(8,17,-9,-18)
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,-15,-3,12)(-2,15)(-4,7,-20,-10,-12)(-5,-19,-7)(-6,19)(-8,-18,5)(-9,20,6,18)(-11,10)(-13,-17,8,4)(-14,1,11,9,17)(-16,13,3)(2,14,16)
Multiloop annotated with half-edges
12^2_174 annotated with half-edges