$12^{1}_{352}$ - Minimal pinning sets
Pinning sets for 12^1_352 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_352 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 184
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97614
on average over minimal pinning sets: 2.38333
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, 3, 5, 10}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 5, 8, 9}
6
[2, 2, 2, 2, 3, 4]
2.50
b (minimal)
• {1, 2, 3, 4, 5, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
c (minimal)
• {1, 2, 3, 5, 9, 12}
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
3
7
2.5
7
0
0
33
2.76
8
0
0
54
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
3
180
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,6,7],[0,8,8,4],[0,3,9,1],[1,9,9,6],[1,5,2,2],[2,9,8,8],[3,7,7,3],[4,7,5,5]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[17,4,18,5],[12,19,13,20],[6,15,7,16],[1,16,2,17],[10,3,11,4],[18,11,19,12],[13,9,14,8],[14,7,15,8],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(15,2,-16,-3)(12,5,-13,-6)(6,11,-7,-12)(20,7,-1,-8)(16,9,-17,-10)(4,13,-5,-14)(19,14,-20,-15)(10,17,-11,-18)(3,18,-4,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,15,-20,-8)(-3,-19,-15)(-4,-14,19)(-5,12,-7,20,14)(-6,-12)(-9,16,2)(-10,-18,3,-16)(-11,6,-13,4,18)(-17,10)(1,7,11,17,9)(5,13)
Loop annotated with half-edges
12^1_352 annotated with half-edges