$12^{1}_{85}$ - Minimal pinning sets
Pinning sets for 12^1_85 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_85 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, 3, 10, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 7, 9, 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
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, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,4,0],[0,4,5,5],[0,6,7,1],[1,8,5,2],[2,4,6,2],[3,5,9,7],[3,6,9,8],[4,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,19,11,20],[8,13,9,14],[1,18,2,19],[11,15,12,14],[12,7,13,8],[17,6,18,7],[2,6,3,5],[15,5,16,4],[16,3,17,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,11,-1,-12)(1,18,-2,-19)(13,2,-14,-3)(8,3,-9,-4)(16,5,-17,-6)(4,9,-5,-10)(10,7,-11,-8)(17,14,-18,-15)(6,15,-7,-16)(12,19,-13,-20)
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,-19,12)(-2,13,19)(-3,8,-11,20,-13)(-4,-10,-8)(-5,16,-7,10)(-6,-16)(-9,4)(-12,-20)(-14,17,5,9,3)(-15,6,-17)(-18,1,11,7,15)(2,18,14)
Loop annotated with half-edges
12^1_85 annotated with half-edges