$12^{1}_{143}$ - Minimal pinning sets
Pinning sets for 12^1_143 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_143 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 192
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: 2.96906
on average over minimal pinning sets: 2.2
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)
• {2, 3, 4, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 6, 9}
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
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
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
2
0
190
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,6,7,7],[0,8,1,0],[1,9,6,5],[1,4,6,6],[2,5,5,4],[2,9,8,2],[3,7,9,9],[4,8,8,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[11,8,12,9],[14,19,15,20],[10,2,11,1],[4,7,5,8],[12,5,13,6],[6,13,7,14],[18,15,19,16],[2,18,3,17],[3,16,4,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,3,-11,-4)(2,5,-3,-6)(9,6,-10,-7)(4,11,-5,-12)(16,13,-17,-14)(1,14,-2,-15)(15,20,-16,-1)(12,17,-13,-18)(7,18,-8,-19)(19,8,-20,-9)
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)(-2,-6,9,-20,15)(-3,10,6)(-4,-12,-18,7,-10)(-5,2,14,-17,12)(-7,-19,-9)(-8,19)(-11,4)(-13,16,20,8,18)(-14,1,-16)(3,5,11)(13,17)
Loop annotated with half-edges
12^1_143 annotated with half-edges