$12^{1}_{264}$ - Minimal pinning sets
Pinning sets for 12^1_264 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_264 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 144
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: 2.97944
on average over minimal pinning sets: 2.375
on average over optimal pinning sets: 2.375
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 6, 7, 12}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 3, 4, 6, 7, 12}
6
[2, 2, 2, 2, 3, 4]
2.50
C (optimal)
• {1, 3, 6, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
D (optimal)
• {1, 3, 6, 7, 11, 12}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
4
0
0
2.38
7
0
0
20
2.68
8
0
0
41
2.9
9
0
0
44
3.06
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
140
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,4,5,3],[0,2,5,6],[1,7,2,1],[2,8,6,3],[3,5,9,9],[4,9,8,8],[5,7,7,9],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[19,6,20,7],[8,18,9,17],[1,17,2,16],[5,18,6,19],[9,3,10,2],[10,15,11,16],[4,13,5,14],[3,13,4,12],[14,11,15,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,7,-1,-8)(17,4,-18,-5)(12,9,-13,-10)(10,1,-11,-2)(2,11,-3,-12)(8,13,-9,-14)(14,19,-15,-20)(15,6,-16,-7)(3,16,-4,-17)(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,-13,8)(-2,-12,-10)(-3,-17,-5,-19,14,-9,12)(-4,17)(-6,15,19)(-7,20,-15)(-8,-14,-20)(-11,2)(-16,3,11,1,7)(-18,5)(4,16,6,18)(9,13)
Loop annotated with half-edges
12^1_264 annotated with half-edges