$12^{2}_{57}$ - Minimal pinning sets
Pinning sets for 12^2_57 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_57 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 368
of which optimal: 5
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10822
on average over minimal pinning sets: 2.64444
on average over optimal pinning sets: 2.64
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 10, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 2, 5, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
D (optimal)
• {1, 2, 4, 8, 9}
5
[2, 2, 3, 3, 4]
2.80
E (optimal)
• {1, 2, 4, 9, 10}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 5, 6, 10, 11}
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
5
5
0
0
2.64
6
0
1
30
2.85
7
0
0
80
3.0
8
0
0
111
3.12
9
0
0
89
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
5
1
362
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,7,8,3],[0,2,8,9],[0,6,5,5],[1,4,4,6],[1,5,4,7],[1,6,9,2],[2,9,9,3],[3,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,3,10,4],[15,7,16,8],[16,19,17,20],[1,11,2,12],[12,2,13,3],[10,13,11,14],[4,14,5,15],[18,6,19,7],[17,6,18,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(18,1,-19,-2)(3,14,-4,-15)(15,4,-16,-5)(12,5,-13,-6)(7,20,-8,-9)(19,10,-20,-11)(2,11,-3,-12)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-7,-9)(-2,-12,-6,-18)(-3,-15,-5,12)(-4,15)(-8,9)(-10,19,1)(-11,2,-19)(-13,-17,6)(-14,3,11,-20,7,17)(-16,13,5)(4,14,16)(8,20,10)
Multiloop annotated with half-edges
12^2_57 annotated with half-edges