$12^{1}_{176}$ - Minimal pinning sets
Pinning sets for 12^1_176 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_176 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 320
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03463
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 5, 11, 12}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
1
8
2.56
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
318
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,5],[0,5,6,4],[1,3,7,8],[2,8,3,2],[3,9,9,7],[4,6,9,8],[4,7,9,5],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,19,11,20],[11,8,12,9],[1,7,2,6],[18,5,19,6],[7,12,8,13],[2,16,3,15],[17,14,18,15],[4,13,5,14],[16,4,17,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,13,-1,-14)(15,2,-16,-3)(3,18,-4,-19)(8,5,-9,-6)(17,6,-18,-7)(4,9,-5,-10)(1,10,-2,-11)(14,11,-15,-12)(12,19,-13,-20)(7,16,-8,-17)
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,-11,14)(-2,15,11)(-3,-19,12,-15)(-4,-10,1,13,19)(-5,8,16,2,10)(-6,17,-8)(-7,-17)(-9,4,18,6)(-12,-20,-14)(-13,20)(-16,7,-18,3)(5,9)
Loop annotated with half-edges
12^1_176 annotated with half-edges