$12^{1}_{36}$ - Minimal pinning sets
Pinning sets for 12^1_36 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_36 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)
• {1, 2, 4, 5, 7, 10}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 4, 5, 7, 12}
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, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,6,3,0],[0,2,6,4],[1,3,7,1],[1,8,8,9],[2,9,9,3],[4,9,8,8],[5,7,7,5],[5,7,6,6]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,8,17,7],[14,19,15,20],[1,19,2,18],[8,18,9,17],[11,6,12,7],[13,2,14,3],[9,4,10,5],[5,10,6,11],[12,4,13,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(1,10,-2,-11)(7,2,-8,-3)(9,4,-10,-5)(5,14,-6,-15)(3,8,-4,-9)(18,13,-19,-14)(15,6,-16,-7)(16,19,-17,-20)(12,17,-13,-18)
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,-11)(-2,7,-16,-20,11)(-3,-9,-5,-15,-7)(-4,9)(-6,15)(-8,3)(-10,1,-12,-18,-14,5)(-13,18)(-17,12,20)(-19,16,6,14)(2,10,4,8)(13,17,19)
Loop annotated with half-edges
12^1_36 annotated with half-edges