$12^{2}_{64}$ - Minimal pinning sets
Pinning sets for 12^2_64 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_64 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 200
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: 3.03442
on average over minimal pinning sets: 2.49286
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 6, 7, 10, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 5, 6, 8, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 2, 5, 6, 7, 10, 11}
7
[2, 2, 2, 3, 3, 3, 3]
2.57
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.4
6
0
2
7
2.63
7
0
1
31
2.82
8
0
0
58
2.99
9
0
0
58
3.12
10
0
0
32
3.22
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
3
196
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,2],[0,1,4,0],[0,5,6,1],[1,7,7,2],[3,7,8,6],[3,5,8,9],[4,9,5,4],[5,9,9,6],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,18,12,19],[19,9,20,10],[1,17,2,18],[12,8,13,9],[16,6,17,7],[2,6,3,5],[7,13,8,14],[15,3,16,4],[4,14,5,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(3,10,-4,-1)(13,4,-14,-5)(19,6,-20,-7)(16,7,-17,-8)(5,20,-6,-11)(2,11,-3,-12)(9,14,-10,-15)(18,15,-19,-16)(8,17,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3)(-2,-12)(-4,13,1)(-5,-11,2,-13)(-6,19,15,-10,3,11)(-7,16,-19)(-8,-18,-16)(-9,-15,18)(-14,9,17,7,-20,5)(-17,8)(4,10,14)(6,20)
Multiloop annotated with half-edges
12^2_64 annotated with half-edges