$12^{1}_{169}$ - Minimal pinning sets
Pinning sets for 12^1_169 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_169 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 256
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96564
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 7}
4
[2, 2, 2, 2]
2.00
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
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 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],[3,8,9,9],[4,9,9,8],[4,7,6,5],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[19,8,20,9],[10,8,11,7],[1,12,2,13],[13,18,14,19],[11,6,12,7],[2,15,3,16],[4,17,5,18],[14,5,15,6],[3,17,4,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,3,-13,-4)(9,4,-10,-5)(5,8,-6,-9)(17,6,-18,-7)(19,10,-20,-11)(20,13,-1,-14)(14,1,-15,-2)(2,15,-3,-16)(11,16,-12,-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)(-2,-16,11,-20,-14)(-3,12,16)(-4,9,-6,17,-12)(-5,-9)(-7,-19,-11,-17)(-8,5,-10,19)(-13,20,10,4)(-15,2)(-18,7)(1,13,3,15)(6,8,18)
Loop annotated with half-edges
12^1_169 annotated with half-edges