$12^{2}_{154}$ - Minimal pinning sets
Pinning sets for 12^2_154 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_154 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 128
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90623
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 6, 7}
5
[2, 2, 2, 2, 2]
2.00
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
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,6,7,7],[0,7,1,0],[1,7,6,5],[1,4,8,8],[2,9,9,4],[2,4,3,2],[5,9,9,5],[6,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,19,6,18],[3,13,4,14],[19,1,20,2],[6,12,7,11],[17,10,18,11],[14,8,15,7],[12,2,13,3],[9,16,10,17],[8,16,9,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-5,-2)(13,2,-14,-3)(18,11,-19,-12)(12,7,-13,-8)(3,14,-4,-15)(16,9,-17,-10)(10,17,-11,-18)(19,6,-20,-7)(4,5,-1,-6)(8,15,-9,-16)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,6)(-2,13,7,-20)(-3,-15,8,-13)(-4,-6,19,11,17,9,15)(-5,4,14,2)(-7,12,-19)(-8,-16,-10,-18,-12)(-9,16)(-11,18)(-14,3)(-17,10)(1,5)
Multiloop annotated with half-edges
12^2_154 annotated with half-edges