$12^{1}_{81}$ - Minimal pinning sets
Pinning sets for 12^1_81
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_81
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)
•
{1, 2, 3, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
•
{1, 2, 3, 5, 7}
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, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,4],[0,5,6,1],[2,7,5,2],[3,4,8,6],[3,5,8,7],[4,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[19,8,20,9],[10,8,11,7],[1,18,2,19],[11,6,12,7],[12,17,13,18],[2,13,3,14],[14,5,15,6],[16,3,17,4],[4,15,5,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(12,3,-13,-4)(9,4,-10,-5)(5,8,-6,-9)(17,6,-18,-7)(20,11,-1,-12)(2,13,-3,-14)(10,15,-11,-16)(19,16,-20,-17)(7,18,-8,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-3,12)(-2,-14)(-4,9,-6,17,-20,-12)(-5,-9)(-7,-19,-17)(-8,5,-10,-16,19)(-11,20,16)(-13,2,-15,10,4)(-18,7)(1,11,15)(3,13)(6,8,18)
Loop annotated with half-edges
12^1_81 annotated with half-edges