$12^{1}_{179}$ - Minimal pinning sets
Pinning sets for 12^1_179 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_179 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 224
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03933
on average over minimal pinning sets: 2.51333
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, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 5, 6, 9, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 5, 6, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 2, 4, 6, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
d (minimal)
• {1, 2, 4, 8, 9, 11}
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
1
0
0
2.4
6
0
4
7
2.62
7
0
0
39
2.84
8
0
0
67
3.01
9
0
0
63
3.14
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
4
219
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],[0,5,5,6],[0,6,7,3],[0,2,7,8],[0,8,9,9],[1,9,8,1],[1,7,7,2],[2,6,6,3],[3,5,9,4],[4,8,5,4]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[4,9,5,10],[19,16,20,17],[6,16,7,15],[1,12,2,13],[10,3,11,4],[8,17,9,18],[18,7,19,8],[11,14,12,15],[2,14,3,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(15,4,-16,-5)(1,6,-2,-7)(18,9,-19,-10)(10,19,-11,-20)(8,11,-9,-12)(12,3,-13,-4)(16,13,-17,-14)(5,14,-6,-15)(2,17,-3,-18)
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,-7)(-2,-18,-10,-20,7)(-3,12,-9,18)(-4,15,-6,1,-8,-12)(-5,-15)(-11,8,20)(-13,16,4)(-14,5,-16)(-17,2,6,14)(-19,10)(3,17,13)(9,11,19)
Loop annotated with half-edges
12^1_179 annotated with half-edges