$12^{1}_{40}$ - Minimal pinning sets
Pinning sets for 12^1_40 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_40 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, 3, 6, 9, 10}
6
[2, 2, 2, 2, 2, 3]
2.17
a (minimal)
• {1, 2, 3, 6, 8, 9, 11}
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, 5, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,7,0],[0,4,4,1],[1,3,3,5],[1,4,7,6],[2,5,8,8],[2,9,9,5],[6,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[12,9,13,10],[19,14,20,15],[1,11,2,10],[2,11,3,12],[3,8,4,9],[15,4,16,5],[7,18,8,19],[16,6,17,5],[17,6,18,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(2,13,-3,-14)(14,3,-15,-4)(20,5,-1,-6)(9,6,-10,-7)(7,18,-8,-19)(19,8,-20,-9)(16,11,-17,-12)(4,15,-5,-16)(10,17,-11,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-17,10,6)(-2,-14,-4,-16,-12)(-3,14)(-5,20,8,18,-11,16)(-6,9,-20)(-7,-19,-9)(-8,19)(-10,-18,7)(-13,2)(-15,4)(1,5,15,3,13)(11,17)
Loop annotated with half-edges
12^1_40 annotated with half-edges