$12^{1}_{21}$ - Minimal pinning sets
Pinning sets for 12^1_21
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_21
Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 320
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: 3.03463
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{2, 3, 5, 7}
4
[2, 2, 2, 3]
2.25
a (minimal)
•
{1, 2, 4, 5, 7}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
1
8
2.56
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
318
Other information about this loop
Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,5,4,3],[0,2,1,0],[1,2,6,6],[1,7,8,2],[4,9,7,4],[5,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[13,10,14,11],[19,2,20,3],[12,2,13,1],[9,18,10,19],[14,4,15,3],[17,8,18,9],[4,8,5,7],[15,7,16,6],[16,5,17,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,11,-1,-12)(13,2,-14,-3)(3,12,-4,-13)(4,1,-5,-2)(14,5,-15,-6)(10,7,-11,-8)(6,15,-7,-16)(8,17,-9,-18)(18,9,-19,-10)(16,19,-17,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,12)(-2,13,-4)(-3,-13)(-5,14,2)(-6,-16,-20,-12,3,-14)(-7,10,-19,16)(-8,-18,-10)(-9,18)(-11,20,-17,8)(-15,6)(1,11,7,15,5)(9,17,19)
Loop annotated with half-edges
12^1_21 annotated with half-edges