$12^{3}_{52}$ - Minimal pinning sets
Pinning sets for 12^3_52 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_52 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 220
of which optimal: 2
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98171
on average over minimal pinning sets: 2.48
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {2, 3, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 7, 10, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
b (minimal)
• {1, 2, 4, 7, 9, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
c (minimal)
• {1, 2, 4, 6, 7, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
3
13
2.56
7
0
0
45
2.81
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
3
215
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,5,6],[0,6,7,4],[1,3,8,9],[1,9,2,2],[2,7,7,3],[3,6,6,8],[4,7,9,9],[4,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[10,14,1,11],[11,9,12,10],[13,20,14,15],[1,4,2,5],[5,8,6,9],[12,16,13,15],[3,19,4,20],[2,19,3,18],[7,17,8,18],[6,17,7,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,2,-10,-3)(3,6,-4,-7)(18,7,-19,-8)(8,17,-9,-18)(15,4,-16,-5)(5,16,-6,-11)(11,10,-12,-1)(1,12,-2,-13)(20,13,-17,-14)(14,19,-15,-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,-13,20,-15,-5,-11)(-2,9,17,13)(-3,-7,18,-9)(-4,15,19,7)(-6,3,-10,11)(-8,-18)(-12,1)(-14,-20)(-16,5)(-17,8,-19,14)(2,12,10)(4,6,16)
Multiloop annotated with half-edges
12^3_52 annotated with half-edges