$12^{1}_{89}$ - Minimal pinning sets
Pinning sets for 12^1_89 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_89 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 352
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: 3.04132
on average over minimal pinning sets: 2.41667
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, 3, 5, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 4, 5, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 4, 5, 10, 11}
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
2
8
2.56
6
0
0
39
2.79
7
0
0
81
2.95
8
0
0
100
3.08
9
0
0
76
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
2
349
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,6],[0,6,7,4],[1,3,7,7],[2,8,9,9],[2,9,8,3],[3,8,4,4],[5,7,6,9],[5,8,6,5]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,19,11,20],[11,8,12,9],[1,5,2,4],[18,3,19,4],[14,7,15,8],[12,6,13,5],[2,17,3,18],[13,16,14,17],[6,15,7,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(1,18,-2,-19)(2,9,-3,-10)(11,4,-12,-5)(8,5,-9,-6)(17,6,-18,-7)(3,12,-4,-13)(10,13,-11,-14)(19,14,-20,-15)(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,-19,-15)(-2,-10,-14,19)(-3,-13,10)(-4,11,13)(-5,8,16,20,14,-11)(-6,17,-8)(-7,-17)(-9,2,18,6)(-12,3,9,5)(-16,7,-18,1)(-20,15)(4,12)
Loop annotated with half-edges
12^1_89 annotated with half-edges