$12^{1}_{275}$ - Minimal pinning sets
Pinning sets for 12^1_275 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_275 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 120
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: 2.91853
on average over minimal pinning sets: 2.25
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 7, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 3, 5, 6, 9, 11}
6
[2, 2, 2, 2, 2, 4]
2.33
C (optimal)
• {1, 2, 3, 6, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
D (optimal)
• {1, 3, 4, 6, 9, 11}
6
[2, 2, 2, 2, 2, 4]
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
2.25
7
0
0
18
2.6
8
0
0
34
2.85
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
4
0
116
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,5],[0,6,7,7],[1,8,8,5],[2,4,6,2],[3,5,8,9],[3,9,9,3],[4,9,6,4],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,19,9,20],[9,6,10,7],[1,14,2,15],[18,11,19,12],[5,10,6,11],[13,4,14,5],[2,16,3,15],[12,17,13,18],[3,16,4,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(6,3,-7,-4)(17,4,-18,-5)(2,7,-3,-8)(15,8,-16,-9)(18,11,-19,-12)(12,19,-13,-20)(20,13,-1,-14)(9,14,-10,-15)(5,16,-6,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,14)(-2,-8,15,-10)(-3,6,16,8)(-4,17,-6)(-5,-17)(-7,2,-11,18,4)(-9,-15)(-12,-20,-14,9,-16,5,-18)(-13,20)(-19,12)(1,13,19,11)(3,7)
Loop annotated with half-edges
12^1_275 annotated with half-edges