$12^{1}_{256}$ - Minimal pinning sets
Pinning sets for 12^1_256 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_256 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 80
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: 2.91429
on average over minimal pinning sets: 2.22619
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 8, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
a (minimal)
• {1, 2, 5, 6, 7, 8, 9}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
1
0
0
2.17
7
0
1
6
2.47
8
0
0
19
2.73
9
0
0
26
2.94
10
0
0
19
3.12
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
1
78
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 8]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,4,0],[0,5,5,0],[1,6,7,4],[1,3,7,5],[2,4,8,2],[3,9,9,7],[3,6,8,4],[5,7,9,9],[6,8,8,6]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[19,8,20,9],[10,2,11,1],[18,13,19,14],[7,12,8,13],[2,12,3,11],[14,5,15,6],[6,17,7,18],[3,17,4,16],[4,15,5,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(16,5,-17,-6)(8,19,-9,-20)(4,9,-5,-10)(15,10,-16,-11)(1,12,-2,-13)(13,20,-14,-1)(3,14,-4,-15)(6,17,-7,-18)(18,7,-19,-8)
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,-13)(-2,11,-16,-6,-18,-8,-20,13)(-3,-15,-11)(-4,-10,15)(-5,16,10)(-7,18)(-9,4,14,20)(-12,1,-14,3)(-17,6)(-19,8)(2,12)(5,9,19,7,17)
Loop annotated with half-edges
12^1_256 annotated with half-edges