$12^{1}_{154}$ - Minimal pinning sets
Pinning sets for 12^1_154 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_154 Pinning data
Pinning number of this loop: 7
Total number of pinning sets: 48
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: 2.86152
on average over minimal pinning sets: 2.14286
on average over optimal pinning sets: 2.14286
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 6, 7, 9}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
B (optimal)
• {1, 2, 4, 5, 6, 7, 12}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
7
2
0
0
2.14
8
0
0
9
2.53
9
0
0
16
2.82
10
0
0
14
3.04
11
0
0
6
3.21
12
0
0
1
3.33
Total
2
0
46
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,5,6,3],[0,2,6,7],[1,8,8,1],[2,8,8,7],[2,9,9,3],[3,9,9,5],[4,5,5,4],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[19,8,20,9],[10,16,11,15],[1,15,2,14],[7,18,8,19],[16,6,17,5],[11,3,12,2],[4,13,5,14],[17,6,18,7],[3,13,4,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,9,-1,-10)(3,14,-4,-15)(15,4,-16,-5)(17,6,-18,-7)(10,1,-11,-2)(2,11,-3,-12)(12,19,-13,-20)(13,8,-14,-9)(5,16,-6,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,-12,-20,-10)(-3,-15,-5,-17,-7,-19,12)(-4,15)(-6,17)(-8,13,19)(-9,20,-13)(-11,2)(-14,3,11,1,9)(-16,5)(-18,7)(4,14,8,18,6,16)
Loop annotated with half-edges
12^1_154 annotated with half-edges