$12^{2}_{141}$ - Minimal pinning sets
Pinning sets for 12^2_141 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_141 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 8, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 3, 4, 5, 8, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,3,4],[0,5,5,0],[0,6,1,1],[1,6,6,7],[2,7,8,2],[3,9,4,4],[4,9,8,5],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[3,16,4,1],[2,20,3,17],[15,4,16,5],[1,18,2,17],[9,19,10,20],[5,14,6,15],[18,8,19,9],[10,13,11,14],[6,11,7,12],[12,7,13,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,2,-14,-3)(10,5,-11,-6)(19,6,-20,-7)(4,9,-5,-10)(8,11,-9,-12)(3,12,-4,-13)(1,14,-2,-15)(7,20,-8,-17)(16,17,-1,-18)(18,15,-19,-16)
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,-15,18)(-2,13,-4,-10,-6,19,15)(-3,-13)(-5,10)(-7,-17,16,-19)(-8,-12,3,-14,1,17)(-9,4,12)(-11,8,20,6)(-16,-18)(-20,7)(2,14)(5,9,11)
Multiloop annotated with half-edges
12^2_141 annotated with half-edges