$12^{1}_{48}$ - Minimal pinning sets
Pinning sets for 12^1_48 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_48 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, 4, 4, 4, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,7],[0,7,8,8],[0,5,5,1],[1,4,4,6],[1,5,9,2],[2,9,3,2],[3,9,9,3],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,13,11,14],[4,19,5,20],[8,17,9,18],[1,12,2,13],[11,2,12,3],[14,3,15,4],[18,5,19,6],[16,7,17,8],[15,7,16,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(15,2,-16,-3)(3,12,-4,-13)(13,4,-14,-5)(17,6,-18,-7)(18,9,-19,-10)(7,10,-8,-11)(5,14,-6,-15)(1,16,-2,-17)(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,-17,-7,-11)(-2,15,-6,17)(-3,-13,-5,-15)(-4,13)(-8,-20,11)(-9,18,6,14,4,12,20)(-10,7,-18)(-12,3,-16,1)(-14,5)(-19,8,10)(2,16)(9,19)
Loop annotated with half-edges
12^1_48 annotated with half-edges