$12^{2}_{199}$ - Minimal pinning sets
Pinning sets for 12^2_199 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_199 Pinning data
Pinning number of this multiloop: 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, 9}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 7, 8, 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 multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,7,3],[0,2,7,7],[0,8,9,9],[1,9,9,6],[1,5,8,8],[2,8,3,3],[4,7,6,6],[4,5,5,4]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,9,16,10],[10,13,11,14],[11,19,12,20],[1,6,2,7],[3,8,4,9],[16,4,17,5],[18,12,19,13],[5,17,6,18],[2,8,3,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(13,4,-14,-5)(5,14,-6,-1)(1,6,-2,-7)(20,7,-15,-8)(17,10,-18,-11)(8,11,-9,-12)(3,16,-4,-17)(9,18,-10,-19)(12,19,-13,-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,-7,20,-13,-5)(-2,15,7)(-3,-17,-11,8,-15)(-4,13,19,-10,17)(-6,1)(-8,-12,-20)(-9,-19,12)(-14,5)(-16,3)(-18,9,11)(2,6,14,4,16)(10,18)
Multiloop annotated with half-edges
12^2_199 annotated with half-edges