$12^{1}_{271}$ - Minimal pinning sets
Pinning sets for 12^1_271 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_271 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, 3, 4, 6, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 4, 7, 9}
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, 3, 4, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,3],[0,2,7,7],[0,8,9,9],[1,6,6,1],[2,5,5,9],[3,8,8,3],[4,7,7,9],[4,8,6,4]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,5,17,6],[6,19,7,20],[7,14,8,15],[1,10,2,11],[4,17,5,18],[18,3,19,4],[13,8,14,9],[9,12,10,13],[2,12,3,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(13,2,-14,-3)(10,5,-11,-6)(18,7,-19,-8)(8,19,-9,-20)(6,9,-7,-10)(4,11,-5,-12)(17,12,-18,-13)(1,14,-2,-15)(3,16,-4,-17)
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)(-2,13,-18,-8,-20,15)(-3,-17,-13)(-4,-12,17)(-5,10,-7,18,12)(-6,-10)(-9,6,-11,4,16,20)(-14,1,-16,3)(-19,8)(2,14)(5,11)(7,9,19)
Loop annotated with half-edges
12^1_271 annotated with half-edges