$12^{1}_{261}$ - Minimal pinning sets
Pinning sets for 12^1_261 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_261 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 180
of which optimal: 4
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0386
on average over minimal pinning sets: 2.57143
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 6, 9, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
B (optimal)
• {1, 3, 6, 7, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
C (optimal)
• {1, 3, 5, 6, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
D (optimal)
• {1, 4, 5, 6, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
a (minimal)
• {1, 3, 4, 6, 7, 9, 10}
7
[2, 2, 2, 3, 3, 3, 3]
2.57
b (minimal)
• {1, 2, 3, 6, 7, 9, 10}
7
[2, 2, 2, 3, 3, 3, 4]
2.71
c (minimal)
• {1, 2, 3, 5, 6, 9, 10}
7
[2, 2, 2, 3, 3, 3, 4]
2.71
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
2.5
7
0
3
21
2.75
8
0
0
53
2.96
9
0
0
57
3.12
10
0
0
32
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
3
173
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,4,5],[0,6,7,7],[1,8,5,2],[2,4,8,6],[3,5,9,7],[3,6,9,3],[4,9,9,5],[6,8,8,7]]
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],[5,18,6,19],[10,18,11,17],[13,16,14,17],[2,16,3,15],[4,11,5,12],[12,3,13,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(6,3,-7,-4)(17,4,-18,-5)(1,8,-2,-9)(18,11,-19,-12)(12,19,-13,-20)(10,13,-11,-14)(14,7,-15,-8)(2,15,-3,-16)(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,-9)(-2,-16,5,-18,-12,-20,9)(-3,6,16)(-4,17,-6)(-5,-17)(-7,14,-11,18,4)(-8,1,-10,-14)(-13,10,20)(-15,2,8)(-19,12)(3,15,7)(11,13,19)
Loop annotated with half-edges
12^1_261 annotated with half-edges