$12^{1}_{43}$ - Minimal pinning sets
Pinning sets for 12^1_43 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_43 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, 5, 6, 9, 10}
6
[2, 2, 2, 2, 2, 3]
2.17
a (minimal)
• {1, 2, 5, 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, 4, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,6,0],[0,4,4,1],[1,3,3,5],[1,4,7,8],[2,8,7,2],[5,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,13,11,14],[8,19,9,20],[1,12,2,13],[11,2,12,3],[14,3,15,4],[18,7,19,8],[15,7,16,6],[4,17,5,18],[16,5,17,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(13,2,-14,-3)(3,12,-4,-13)(15,4,-16,-5)(17,6,-18,-7)(18,9,-19,-10)(7,10,-8,-11)(1,14,-2,-15)(5,16,-6,-17)(8,19,-9,-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,-15,-5,-17,-7,-11)(-2,13,-4,15)(-3,-13)(-6,17)(-8,-20,11)(-9,18,6,16,4,12,20)(-10,7,-18)(-12,3,-14,1)(-16,5)(-19,8,10)(2,14)(9,19)
Loop annotated with half-edges
12^1_43 annotated with half-edges