$12^{1}_{273}$ - Minimal pinning sets
Pinning sets for 12^1_273 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_273 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, 5, 7, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
a (minimal)
• {1, 2, 3, 5, 6, 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, 4, 5, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,5,6],[1,7,7,8],[2,8,8,3],[3,8,9,9],[4,9,9,4],[4,6,5,5],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[20,11,1,12],[12,19,13,20],[13,10,14,11],[1,14,2,15],[7,18,8,19],[9,2,10,3],[15,4,16,5],[17,6,18,7],[8,4,9,3],[16,6,17,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,5,-1,-6)(12,1,-13,-2)(2,13,-3,-14)(14,3,-15,-4)(10,7,-11,-8)(17,8,-18,-9)(18,11,-19,-12)(4,15,-5,-16)(9,16,-10,-17)(6,19,-7,-20)
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,-19,6)(-2,-14,-4,-16,9,-18,-12)(-3,14)(-5,20,-7,10,16)(-6,-20)(-8,17,-10)(-9,-17)(-11,18,8)(-13,2)(-15,4)(1,5,15,3,13)(7,19,11)
Loop annotated with half-edges
12^1_273 annotated with half-edges