$12^{2}_{41}$ - Minimal pinning sets
Pinning sets for 12^2_41 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_41 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 176
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97631
on average over minimal pinning sets: 2.34444
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)
• {1, 2, 7, 8, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 6, 7, 11, 12}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 6, 7, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
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
2.2
6
0
2
7
2.5
7
0
0
30
2.75
8
0
0
51
2.94
9
0
0
49
3.08
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
2
173
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: [[1,2,2,3],[0,4,5,5],[0,6,3,0],[0,2,6,4],[1,3,7,5],[1,4,8,1],[2,8,7,3],[4,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,7,14,8],[19,11,20,12],[1,11,2,10],[6,9,7,10],[14,9,15,8],[18,2,19,3],[5,17,6,18],[15,4,16,3],[16,4,17,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,13,-1,-14)(14,1,-15,-2)(19,4,-20,-5)(5,20,-6,-13)(3,6,-4,-7)(7,10,-8,-11)(17,8,-18,-9)(16,11,-17,-12)(2,15,-3,-16)(9,18,-10,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14)(-2,-16,-12,-14)(-3,-7,-11,16)(-4,19,-10,7)(-5,-13,12,-17,-9,-19)(-6,3,15,1,13)(-8,17,11)(-15,2)(-18,9)(-20,5)(4,6,20)(8,10,18)
Multiloop annotated with half-edges
12^2_41 annotated with half-edges