$12^{2}_{237}$ - Minimal pinning sets
Pinning sets for 12^2_237 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_237 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, 5, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 5, 10, 11}
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],[0,5,6,6],[0,6,3,3],[0,2,2,7],[0,8,9,9],[1,9,8,7],[1,7,2,1],[3,6,5,8],[4,7,5,9],[4,8,5,4]]
PD code (use to draw this multiloop with SnapPy): [[9,12,10,1],[8,20,9,13],[11,6,12,7],[10,6,11,5],[1,15,2,16],[13,3,14,4],[19,7,20,8],[4,18,5,19],[14,17,15,18],[2,17,3,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-19,-2)(3,8,-4,-9)(9,4,-10,-5)(16,5,-17,-6)(7,10,-8,-11)(11,20,-12,-13)(13,12,-14,-1)(19,14,-20,-15)(2,15,-3,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-7,-11,-13)(-2,-16,-6,-18)(-3,-9,-5,16)(-4,9)(-8,3,15,-20,11)(-10,7,17,5)(-12,13)(-14,19,1)(-15,2,-19)(-17,6)(4,8,10)(12,20,14)
Multiloop annotated with half-edges
12^2_237 annotated with half-edges