$12^{1}_{165}$ - Minimal pinning sets
Pinning sets for 12^1_165
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_165
Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
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)
•
{2, 3, 5, 8, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
•
{2, 3, 5, 7, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
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: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,6,6,0],[1,7,2,1],[2,7,8,8],[3,9,7,3],[4,6,9,5],[5,9,9,5],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,18,11,17],[19,16,20,17],[8,1,9,2],[18,12,19,11],[4,15,5,16],[2,7,3,8],[12,3,13,4],[14,5,15,6],[6,13,7,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(12,3,-13,-4)(20,5,-1,-6)(18,7,-19,-8)(9,16,-10,-17)(4,11,-5,-12)(2,13,-3,-14)(17,14,-18,-15)(15,8,-16,-9)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,16,8,-19,6)(-2,-14,17,-10)(-3,12,-5,20,-7,18,14)(-4,-12)(-6,-20)(-8,15,-18)(-9,-17,-15)(-11,4,-13,2)(-16,9)(1,5,11)(3,13)(7,19)
Loop annotated with half-edges
12^1_165 annotated with half-edges