$12^{2}_{63}$ - Minimal pinning sets
Pinning sets for 12^2_63 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_63 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 400
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10335
on average over minimal pinning sets: 2.59167
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 11}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 3, 4, 5, 11}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 2, 3, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 2, 3, 4, 5, 9}
6
[2, 2, 3, 3, 3, 3]
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
2.5
5
0
2
8
2.72
6
0
1
40
2.89
7
0
0
90
3.02
8
0
0
116
3.12
9
0
0
90
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
3
396
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,8,6,3],[0,2,5,4],[0,3,5,5],[1,4,4,3],[1,2,9,7],[1,6,9,8],[2,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,3,12,4],[15,9,16,10],[16,19,17,20],[1,17,2,18],[18,2,19,3],[12,9,13,8],[4,8,5,7],[14,6,15,7],[13,6,14,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,20,-2,-11)(11,2,-12,-3)(3,10,-4,-1)(4,19,-5,-20)(12,5,-13,-6)(15,8,-16,-9)(18,9,-19,-10)(7,14,-8,-15)(13,16,-14,-17)(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,-11,-3)(-2,11)(-4,-20,1)(-5,12,2,20)(-6,-18,-10,3,-12)(-7,-15,-9,18)(-8,15)(-13,-17,6)(-14,7,17)(-16,13,5,19,9)(-19,4,10)(8,14,16)
Multiloop annotated with half-edges
12^2_63 annotated with half-edges