$12^{1}_{211}$ - Minimal pinning sets
Pinning sets for 12^1_211 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_211 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 264
of which optimal: 2
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04089
on average over minimal pinning sets: 2.52222
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, 3, 5, 6, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 5, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 4, 5, 6, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 3, 4, 5, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 2, 3, 5, 6, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
d (minimal)
• {1, 2, 3, 5, 8, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.4
6
0
4
13
2.66
7
0
0
52
2.87
8
0
0
80
3.04
9
0
0
69
3.15
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
4
258
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,7,8,4],[0,3,8,5],[1,4,2,1],[2,9,7,7],[3,6,6,9],[3,9,9,4],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,18,11,17],[19,16,20,17],[8,13,9,14],[1,13,2,12],[18,12,19,11],[4,15,5,16],[14,5,15,6],[7,2,8,3],[3,6,4,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,20,-6,-1)(12,3,-13,-4)(1,4,-2,-5)(18,7,-19,-8)(9,16,-10,-17)(10,19,-11,-20)(6,11,-7,-12)(2,13,-3,-14)(17,14,-18,-15)(15,8,-16,-9)
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,-5)(-2,-14,17,-10,-20,5)(-3,12,-7,18,14)(-4,1,-6,-12)(-8,15,-18)(-9,-17,-15)(-11,6,20)(-13,2,4)(-16,9)(-19,10,16,8)(3,13)(7,11,19)
Loop annotated with half-edges
12^1_211 annotated with half-edges