$12^{1}_{37}$ - Minimal pinning sets
Pinning sets for 12^1_37 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_37 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 7, 10}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 4, 5, 7, 12}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,6,3,0],[0,2,6,7],[1,7,7,1],[1,8,8,6],[2,5,9,3],[3,9,4,4],[5,9,9,5],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[8,15,9,16],[19,10,20,11],[1,19,2,18],[16,7,17,8],[14,3,15,4],[11,3,12,2],[6,17,7,18],[4,13,5,14],[12,5,13,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,9,-1,-10)(12,1,-13,-2)(10,3,-11,-4)(14,5,-15,-6)(17,8,-18,-9)(2,11,-3,-12)(6,13,-7,-14)(4,15,-5,-16)(16,19,-17,-20)(7,18,-8,-19)
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,-3,10)(-2,-12)(-4,-16,-20,-10)(-5,14,-7,-19,16)(-6,-14)(-8,17,19)(-9,20,-17)(-11,2,-13,6,-15,4)(-18,7,13,1,9)(3,11)(5,15)(8,18)
Loop annotated with half-edges
12^1_37 annotated with half-edges