$12^{1}_{274}$ - Minimal pinning sets
Pinning sets for 12^1_274 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_274 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 80
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91429
on average over minimal pinning sets: 2.22619
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)
• {2, 3, 5, 6, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
a (minimal)
• {1, 2, 3, 5, 6, 9, 10}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
1
0
0
2.17
7
0
1
6
2.47
8
0
0
19
2.73
9
0
0
26
2.94
10
0
0
19
3.12
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
1
78
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,6],[0,7,7,8],[0,8,8,0],[1,9,9,5],[1,4,9,6],[1,5,7,7],[2,6,6,2],[2,9,3,3],[4,8,5,4]]
PD code (use to draw this loop with SnapPy): [[20,11,1,12],[12,5,13,6],[8,19,9,20],[10,1,11,2],[15,4,16,5],[13,16,14,17],[6,17,7,18],[18,7,19,8],[9,3,10,2],[3,14,4,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(13,2,-14,-3)(15,6,-16,-7)(18,9,-19,-10)(8,11,-9,-12)(1,12,-2,-13)(3,14,-4,-15)(5,16,-6,-17)(17,4,-18,-5)(10,19,-11,-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,-13,-3,-15,-7)(-2,13)(-4,17,-6,15)(-5,-17)(-8,-12,1)(-9,18,4,14,2,12)(-10,-20,7,-16,5,-18)(-11,8,20)(-14,3)(-19,10)(6,16)(9,11,19)
Loop annotated with half-edges
12^1_274 annotated with half-edges