$12^{1}_{277}$ - Minimal pinning sets
Pinning sets for 12^1_277 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_277 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 10, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 5, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,5,5,3],[0,2,6,0],[1,7,7,1],[1,8,2,2],[3,8,9,9],[4,9,8,4],[5,7,9,6],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[17,10,18,11],[19,2,20,3],[12,2,13,1],[9,16,10,17],[18,4,19,3],[13,7,14,6],[15,8,16,9],[4,8,5,7],[14,5,15,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,7,-1,-8)(1,18,-2,-19)(9,2,-10,-3)(11,4,-12,-5)(3,10,-4,-11)(5,14,-6,-15)(15,6,-16,-7)(13,16,-14,-17)(17,12,-18,-13)(8,19,-9,-20)
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,-19,8)(-2,9,19)(-3,-11,-5,-15,-7,20,-9)(-4,11)(-6,15)(-8,-20)(-10,3)(-12,17,-14,5)(-13,-17)(-16,13,-18,1,7)(2,18,12,4,10)(6,14,16)
Loop annotated with half-edges
12^1_277 annotated with half-edges