$12^{1}_{144}$ - Minimal pinning sets
Pinning sets for 12^1_144
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_144
Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 288
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03466
on average over minimal pinning sets: 2.4
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 2, 4, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
•
{1, 3, 4, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
•
{1, 2, 4, 5, 9}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
•
{1, 3, 4, 5, 9}
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
5
4
0
0
2.4
6
0
0
24
2.69
7
0
0
61
2.9
8
0
0
85
3.05
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
0
284
Other information about this loop
Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,6,7,8],[0,8,1,0],[1,9,6,5],[1,4,6,6],[2,5,5,4],[2,9,9,8],[2,7,9,3],[4,8,7,7]]
PD code (use to draw this loop with SnapPy): [[20,11,1,12],[12,10,13,9],[19,6,20,7],[10,1,11,2],[13,17,14,16],[8,15,9,16],[7,15,8,14],[18,3,19,4],[5,2,6,3],[17,5,18,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,20,-4,-1)(16,1,-17,-2)(2,15,-3,-16)(17,4,-18,-5)(7,10,-8,-11)(18,9,-19,-10)(12,5,-13,-6)(6,13,-7,-14)(14,11,-15,-12)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-3)(-2,-16)(-4,17,1)(-5,12,-15,2,-17)(-6,-14,-12)(-7,-11,14)(-8,-20,3,15,11)(-9,18,4,20)(-10,7,13,5,-18)(-13,6)(-19,8,10)(9,19)
Loop annotated with half-edges
12^1_144 annotated with half-edges