$12^{2}_{116}$ - Minimal pinning sets
Pinning sets for 12^2_116 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_116 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
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: 2.96564
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, 6, 9}
4
[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
4
1
0
0
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,7,7],[0,7,7,6],[0,5,1,1],[1,4,8,9],[2,9,8,3],[2,3,3,2],[5,6,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[3,14,4,1],[2,20,3,15],[13,6,14,7],[4,12,5,11],[1,16,2,15],[16,19,17,20],[7,10,8,11],[5,12,6,13],[8,18,9,19],[17,9,18,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,4,-12,-5)(18,7,-19,-8)(5,8,-6,-9)(9,2,-10,-3)(3,10,-4,-11)(1,12,-2,-13)(6,19,-7,-20)(17,20,-18,-15)(14,15,-1,-16)(16,13,-17,-14)
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,-13,16)(-2,9,-6,-20,17,13)(-3,-11,-5,-9)(-4,11)(-7,18,20)(-8,5,-12,1,15,-18)(-10,3)(-14,-16)(-15,14,-17)(-19,6,8)(2,12,4,10)(7,19)
Multiloop annotated with half-edges
12^2_116 annotated with half-edges