$12^{2}_{75}$ - Minimal pinning sets
Pinning sets for 12^2_75 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_75 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 224
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9785
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 7}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {2, 4, 5, 7, 12}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {2, 4, 5, 7, 10}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.27
6
0
0
18
2.59
7
0
0
46
2.82
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
3
0
221
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,5,6,1],[2,7,8,8],[2,9,6,3],[3,5,7,7],[4,6,6,9],[4,9,9,4],[5,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,9,12,10],[12,19,13,20],[1,8,2,9],[4,18,5,19],[13,7,14,8],[2,14,3,15],[15,3,16,4],[17,5,18,6],[6,16,7,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(14,3,-15,-4)(20,5,-11,-6)(9,6,-10,-7)(18,7,-19,-8)(10,11,-1,-12)(4,13,-5,-14)(2,15,-3,-16)(19,16,-20,-17)(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)(-2,-16,19,7,-10,-12)(-3,14,-5,20,16)(-4,-14)(-6,9,17,-20)(-7,18,-9)(-8,-18)(-11,10,6)(-13,4,-15,2)(-17,8,-19)(1,11,5,13)(3,15)
Multiloop annotated with half-edges
12^2_75 annotated with half-edges