$12^{2}_{49}$ - Minimal pinning sets
Pinning sets for 12^2_49 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_49 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 392
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0598
on average over minimal pinning sets: 2.47333
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)
• {2, 4, 6, 7}
4
[2, 2, 2, 4]
2.50
a (minimal)
• {1, 2, 5, 6, 7}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {2, 5, 6, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
• {2, 3, 6, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
d (minimal)
• {1, 2, 3, 6, 7, 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
2.5
5
0
3
8
2.65
6
0
1
44
2.83
7
0
0
94
2.98
8
0
0
113
3.1
9
0
0
82
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
4
387
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,7,8],[0,5,1,1],[1,4,6,2],[2,5,8,7],[3,6,9,9],[3,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,20,3,11],[13,9,14,10],[4,14,5,15],[1,12,2,11],[12,19,13,20],[8,18,9,19],[5,18,6,17],[15,7,16,8],[6,16,7,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,5,-17,-6)(14,7,-15,-8)(1,8,-2,-9)(2,15,-3,-16)(4,17,-5,-18)(18,3,-19,-4)(6,19,-7,-20)(13,20,-14,-11)(10,11,-1,-12)(12,9,-13,-10)
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,-9,12)(-2,-16,-6,-20,13,9)(-3,18,-5,16)(-4,-18)(-7,14,20)(-8,1,11,-14)(-10,-12)(-11,10,-13)(-15,2,8)(-17,4,-19,6)(3,15,7,19)(5,17)
Multiloop annotated with half-edges
12^2_49 annotated with half-edges