$12^{1}_{189}$ - Minimal pinning sets
Pinning sets for 12^1_189 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_189 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 240
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: 2.97778
on average over minimal pinning sets: 2.3
on average over optimal pinning sets: 2.3
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
B (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 5, 9, 11}
5
[2, 2, 2, 2, 4]
2.40
D (optimal)
• {1, 2, 5, 11, 12}
5
[2, 2, 2, 2, 3]
2.20
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
2.3
6
0
0
22
2.62
7
0
0
52
2.84
8
0
0
69
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
236
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,6],[0,7,7,8],[0,8,8,9],[0,9,5,5],[1,4,4,9],[1,9,7,1],[2,6,8,2],[2,7,3,3],[3,6,5,4]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,3,17,4],[12,19,13,20],[14,9,15,10],[1,6,2,7],[7,2,8,3],[17,5,18,4],[18,11,19,12],[13,11,14,10],[5,8,6,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,20,-6,-1)(14,1,-15,-2)(2,15,-3,-16)(19,4,-20,-5)(3,6,-4,-7)(12,7,-13,-8)(8,11,-9,-12)(16,9,-17,-10)(18,13,-19,-14)(10,17,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-19,-5)(-2,-16,-10,-18,-14)(-3,-7,12,-9,16)(-4,19,13,7)(-6,3,15,1)(-8,-12)(-11,8,-13,18)(-15,2)(-17,10)(-20,5)(4,6,20)(9,11,17)
Loop annotated with half-edges
12^1_189 annotated with half-edges