$12^{1}_{251}$ - Minimal pinning sets
Pinning sets for 12^1_251 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_251 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 288
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: 3.03466
on average over minimal pinning sets: 2.4
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 7}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 4, 7, 12}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {2, 3, 4, 5, 7}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {2, 3, 4, 7, 12}
5
[2, 2, 2, 3, 3]
2.40
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
2.4
6
0
0
24
2.69
7
0
0
61
2.9
8
0
0
85
3.05
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
0
284
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,5,6],[0,6,7,0],[0,7,8,4],[1,3,5,5],[1,4,4,6],[1,5,9,2],[2,9,8,3],[3,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[8,5,9,6],[19,10,20,11],[1,15,2,14],[6,14,7,13],[7,12,8,13],[4,11,5,12],[18,15,19,16],[2,18,3,17],[3,16,4,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,3,-7,-4)(17,8,-18,-9)(2,9,-3,-10)(5,10,-6,-11)(11,4,-12,-5)(12,19,-13,-20)(16,13,-17,-14)(1,14,-2,-15)(15,20,-16,-1)(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,-15)(-2,-10,5,-12,-20,15)(-3,6,10)(-4,11,-6)(-5,-11)(-7,-19,12,4)(-8,17,13,19)(-9,2,14,-17)(-13,16,20)(-14,1,-16)(-18,7,3,9)(8,18)
Loop annotated with half-edges
12^1_251 annotated with half-edges