$12^{2}_{175}$ - Minimal pinning sets
Pinning sets for 12^2_175 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_175 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 380
of which optimal: 1
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03858
on average over minimal pinning sets: 2.50833
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 5, 6, 8, 11}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 5, 6, 11, 12}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
• {1, 5, 6, 10, 11}
5
[2, 2, 2, 3, 4]
2.60
d (minimal)
• {1, 2, 5, 6, 11}
5
[2, 2, 2, 3, 4]
2.60
e (minimal)
• {1, 3, 5, 6, 11}
5
[2, 2, 2, 3, 4]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
5
8
2.57
6
0
0
48
2.81
7
0
0
91
2.97
8
0
0
105
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
5
374
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 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],[0,8,9,5],[1,4,2,1],[2,9,9,3],[3,9,8,8],[3,7,7,4],[4,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,6,10,5],[7,4,8,5],[16,19,17,20],[1,12,2,11],[6,11,7,10],[15,3,16,4],[18,13,19,14],[17,13,18,12],[2,14,3,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(5,2,-6,-3)(3,16,-4,-17)(17,4,-18,-5)(18,7,-19,-8)(19,12,-20,-13)(13,20,-14,-9)(8,9,-1,-10)(11,14,-12,-15)(6,15,-7,-16)
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)(-2,5,-18,-8,-10)(-3,-17,-5)(-4,17)(-6,-16,3)(-7,18,4,16)(-9,8,-19,-13)(-11,-15,6,2)(-12,19,7,15)(-14,11,1,9)(-20,13)(12,14,20)
Multiloop annotated with half-edges
12^2_175 annotated with half-edges