$12^{2}_{55}$ - Minimal pinning sets
Pinning sets for 12^2_55 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_55 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 384
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03466
on average over minimal pinning sets: 2.25
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)
• {2, 4, 9, 10}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {2, 4, 8, 9}
4
[2, 2, 2, 3]
2.25
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.25
5
0
0
15
2.59
6
0
0
49
2.81
7
0
0
91
2.97
8
0
0
105
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
0
382
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,3,3],[0,2,2,7],[0,5,1,1],[1,4,8,6],[2,5,9,7],[3,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[3,6,4,1],[2,20,3,7],[5,15,6,16],[4,15,5,14],[1,8,2,7],[8,19,9,20],[16,9,17,10],[10,13,11,14],[11,18,12,19],[17,12,18,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,4,-2,-5)(17,10,-18,-11)(18,13,-19,-14)(11,14,-12,-15)(15,2,-16,-3)(3,16,-4,-17)(12,19,-13,-20)(9,20,-10,-7)(6,7,-1,-8)(8,5,-9,-6)
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,-5,8)(-2,15,-12,-20,9,5)(-3,-17,-11,-15)(-4,1,7,-10,17)(-6,-8)(-7,6,-9)(-13,18,10,20)(-14,11,-18)(-16,3)(-19,12,14)(2,4,16)(13,19)
Multiloop annotated with half-edges
12^2_55 annotated with half-edges