$12^{1}_{57}$ - Minimal pinning sets
Pinning sets for 12^1_57 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_57 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 160
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: 2.97043
on average over minimal pinning sets: 2.26667
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, 4, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 5, 6, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,3,4],[0,5,6,0],[0,4,1,1],[1,3,7,7],[2,7,8,6],[2,5,8,9],[4,9,5,4],[5,9,9,6],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[11,2,12,3],[19,4,20,5],[1,10,2,11],[12,10,13,9],[5,14,6,15],[15,18,16,19],[13,8,14,9],[6,17,7,18],[16,7,17,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(17,2,-18,-3)(14,5,-15,-6)(11,6,-12,-7)(7,10,-8,-11)(19,8,-20,-9)(3,12,-4,-13)(4,15,-5,-16)(13,16,-14,-17)(1,18,-2,-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,-19,-9)(-2,17,-14,-6,11,-8,19)(-3,-13,-17)(-4,-16,13)(-5,14,16)(-7,-11)(-10,7,-12,3,-18,1)(-15,4,12,6)(-20,9)(2,18)(5,15)(8,10,20)
Loop annotated with half-edges
12^1_57 annotated with half-edges