$12^{1}_{49}$ - Minimal pinning sets
Pinning sets for 12^1_49 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_49 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 160
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: 2.97043
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.2
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, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 8, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
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
2.2
6
0
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,7,8,8],[0,8,9,9],[0,5,5,1],[1,4,4,6],[1,5,7,7],[2,6,6,9],[2,9,3,2],[3,8,7,3]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,11,9,12],[14,19,15,20],[6,17,7,18],[1,10,2,11],[9,2,10,3],[12,3,13,4],[4,13,5,14],[18,15,19,16],[16,5,17,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(13,2,-14,-3)(17,4,-18,-5)(18,7,-19,-8)(5,8,-6,-9)(1,10,-2,-11)(15,12,-16,-13)(3,14,-4,-15)(11,16,-12,-17)(6,19,-7,-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,-11,-17,-5,-9)(-2,13,-16,11)(-3,-15,-13)(-4,17,-12,15)(-6,-20,9)(-7,18,4,14,2,10,20)(-8,5,-18)(-10,1)(-14,3)(-19,6,8)(7,19)(12,16)
Loop annotated with half-edges
12^1_49 annotated with half-edges