$12^{1}_{184}$ - Minimal pinning sets
Pinning sets for 12^1_184 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_184 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 144
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: 2.97193
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 4, 5, 7, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 3, 5, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
D (optimal)
• {1, 2, 3, 5, 7, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
4
0
0
2.33
7
0
0
20
2.66
8
0
0
41
2.89
9
0
0
44
3.06
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
140
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,6,7,0],[1,7,2,1],[2,8,9,6],[3,5,9,9],[3,8,8,4],[5,7,7,9],[5,8,6,6]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,18,9,17],[19,16,20,17],[6,1,7,2],[18,10,19,9],[12,15,13,16],[2,13,3,14],[5,10,6,11],[11,4,12,5],[14,3,15,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(9,2,-10,-3)(18,5,-19,-6)(7,16,-8,-17)(3,8,-4,-9)(20,11,-1,-12)(10,13,-11,-14)(17,14,-18,-15)(15,6,-16,-7)(4,19,-5,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,9,-4,-20,-12)(-3,-9)(-5,18,14,-11,20)(-6,15,-18)(-7,-17,-15)(-8,3,-10,-14,17)(-13,10,2)(-16,7)(-19,4,8,16,6)(1,11,13)(5,19)
Loop annotated with half-edges
12^1_184 annotated with half-edges