$12^{2}_{179}$ - Minimal pinning sets
Pinning sets for 12^2_179 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_179 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 320
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: 3.03463
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 7}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 6, 7, 12}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
1
8
2.56
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
318
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,7],[0,8,8,9],[0,9,7,6],[1,6,6,1],[2,5,5,4],[2,4,9,8],[3,7,9,3],[3,8,7,4]]
PD code (use to draw this multiloop with SnapPy): [[10,5,1,6],[6,11,7,20],[9,19,10,20],[16,4,17,5],[1,14,2,13],[11,8,12,7],[12,8,13,9],[2,18,3,19],[3,15,4,16],[17,15,18,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(7,2,-8,-3)(4,17,-5,-18)(18,5,-19,-6)(19,8,-20,-9)(20,13,-11,-14)(10,11,-1,-12)(9,14,-10,-15)(6,15,-7,-16)(16,3,-17,-4)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,7,15,-10,-12)(-3,16,-7)(-4,-18,-6,-16)(-5,18)(-8,19,5,17,3)(-9,-15,6,-19)(-11,10,14)(-13,20,8,2)(-14,9,-20)(-17,4)(1,11,13)
Multiloop annotated with half-edges
12^2_179 annotated with half-edges