$12^{1}_{237}$ - Minimal pinning sets
Pinning sets for 12^1_237
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_237
Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 312
of which optimal: 4
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04935
on average over minimal pinning sets: 2.55
on average over optimal pinning sets: 2.45
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 2, 3, 5, 7}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
•
{1, 2, 5, 7, 12}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
•
{1, 2, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
•
{1, 2, 5, 6, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
•
{1, 2, 3, 5, 6, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
b (minimal)
•
{1, 2, 5, 6, 9, 12}
6
[2, 2, 2, 3, 4, 4]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.45
6
0
2
24
2.72
7
0
0
68
2.92
8
0
0
94
3.07
9
0
0
75
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
2
306
Other information about this loop
Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,7,7,3],[0,2,7,8],[0,8,9,5],[1,4,6,1],[1,5,9,7],[2,6,3,2],[3,9,9,4],[4,8,8,6]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[6,15,7,16],[10,19,11,20],[8,11,9,12],[1,4,2,5],[16,5,17,6],[17,14,18,15],[18,9,19,10],[12,3,13,4],[2,13,3,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(19,2,-20,-3)(14,3,-15,-4)(16,7,-17,-8)(6,9,-7,-10)(20,11,-1,-12)(15,12,-16,-13)(4,13,-5,-14)(8,17,-9,-18)(5,18,-6,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-7,16,12)(-2,19,-6,-10)(-3,14,-5,-19)(-4,-14)(-8,-18,5,13,-16)(-9,6,18)(-11,20,2)(-12,15,3,-20)(-13,4,-15)(-17,8)(1,11)(7,9,17)
Loop annotated with half-edges
12^1_237 annotated with half-edges