$12^{1}_{39}$ - Minimal pinning sets
Pinning sets for 12^1_39 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_39 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, 6, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
a (minimal)
• {1, 2, 3, 4, 5, 7, 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, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,7],[0,8,8,6],[0,5,5,1],[1,4,4,6],[1,5,3,2],[2,9,9,2],[3,9,9,3],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,13,11,14],[4,19,5,20],[8,15,9,16],[1,12,2,13],[11,2,12,3],[14,3,15,4],[18,5,19,6],[16,7,17,8],[6,17,7,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,20,-14,-1)(11,2,-12,-3)(15,4,-16,-5)(16,7,-17,-8)(5,8,-6,-9)(3,10,-4,-11)(1,12,-2,-13)(19,14,-20,-15)(6,17,-7,-18)(9,18,-10,-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,-13)(-2,11,-4,15,-20,13)(-3,-11)(-5,-9,-19,-15)(-6,-18,9)(-7,16,4,10,18)(-8,5,-16)(-10,3,-12,1,-14,19)(-17,6,8)(2,12)(7,17)(14,20)
Loop annotated with half-edges
12^1_39 annotated with half-edges