$12^{1}_{67}$ - Minimal pinning sets
Pinning sets for 12^1_67 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_67 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, 4, 6, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 6, 7, 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, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,4,5,6],[0,7,7,4],[1,3,2,1],[2,8,9,9],[2,9,8,7],[3,6,8,3],[5,7,6,9],[5,8,6,5]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,19,9,20],[6,17,7,18],[1,11,2,10],[18,9,19,10],[5,14,6,15],[16,3,17,4],[11,3,12,2],[15,12,16,13],[13,4,14,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,9,-1,-10)(14,1,-15,-2)(11,2,-12,-3)(18,5,-19,-6)(16,7,-17,-8)(8,19,-9,-20)(13,10,-14,-11)(3,12,-4,-13)(4,15,-5,-16)(6,17,-7,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,10)(-2,11,-14)(-3,-13,-11)(-4,-16,-8,-20,-10,13)(-5,18,-7,16)(-6,-18)(-9,20)(-12,3)(-15,4,12,2)(-17,6,-19,8)(1,9,19,5,15)(7,17)
Loop annotated with half-edges
12^1_67 annotated with half-edges