$12^{2}_{312}$ - Minimal pinning sets
Pinning sets for 12^2_312 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_312 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 304
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03816
on average over minimal pinning sets: 2.47222
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, 6, 8, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 4, 7, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 4, 5, 7, 8, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
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
0
8
2.58
6
0
2
28
2.78
7
0
0
65
2.93
8
0
0
86
3.06
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
2
301
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: [[1,1,2,3],[0,4,4,0],[0,5,6,3],[0,2,6,7],[1,7,5,1],[2,4,8,9],[2,9,7,3],[3,6,8,4],[5,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,13,16,14],[19,7,20,8],[1,7,2,6],[12,16,13,17],[8,12,9,11],[18,2,19,3],[5,17,6,18],[9,5,10,4],[10,3,11,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,14,-16,-1)(8,1,-9,-2)(2,7,-3,-8)(4,13,-5,-14)(16,5,-17,-6)(9,6,-10,-7)(18,11,-19,-12)(3,20,-4,-15)(12,17,-13,-18)(10,19,-11,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-3,-15)(-2,-8)(-4,-14,15)(-5,16,14)(-6,9,1,-16)(-7,2,-9)(-10,-20,3,7)(-11,18,-13,4,20)(-12,-18)(-17,12,-19,10,6)(5,13,17)(11,19)
Multiloop annotated with half-edges
12^2_312 annotated with half-edges