$12^{1}_{350}$ - Minimal pinning sets
Pinning sets for 12^1_350 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_350 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, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 5, 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, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,7,7,8],[0,8,8,1],[1,9,9,2],[2,9,7,7],[3,6,6,3],[3,9,4,4],[5,8,6,5]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[12,7,13,8],[19,6,20,7],[14,4,15,3],[1,9,2,8],[18,11,19,12],[5,16,6,17],[4,16,5,15],[2,9,3,10],[10,17,11,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,12,-2,-13)(17,4,-18,-5)(10,5,-11,-6)(6,19,-7,-20)(20,7,-1,-8)(16,9,-17,-10)(13,2,-14,-3)(3,14,-4,-15)(8,15,-9,-16)(11,18,-12,-19)
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,-15,8)(-2,13)(-4,17,9,15)(-5,10,-17)(-6,-20,-8,-16,-10)(-7,20)(-9,16)(-11,-19,6)(-12,1,7,19)(-14,3)(-18,11,5)(2,12,18,4,14)
Loop annotated with half-edges
12^1_350 annotated with half-edges