$12^{1}_{164}$ - Minimal pinning sets
Pinning sets for 12^1_164 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_164 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, 3, 4, 8, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 4, 6, 11}
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, 3, 4, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,7,7,4],[0,3,8,8],[1,9,2,1],[2,9,7,7],[3,6,6,3],[4,9,9,4],[5,8,8,6]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,18,9,17],[19,16,20,17],[6,13,7,14],[1,13,2,12],[18,10,19,9],[4,15,5,16],[14,5,15,6],[2,11,3,12],[10,3,11,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(12,3,-13,-4)(18,5,-19,-6)(7,16,-8,-17)(20,9,-1,-10)(10,19,-11,-20)(4,11,-5,-12)(2,13,-3,-14)(17,14,-18,-15)(15,6,-16,-7)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,16,6,-19,10)(-2,-14,17,-8)(-3,12,-5,18,14)(-4,-12)(-6,15,-18)(-7,-17,-15)(-9,20,-11,4,-13,2)(-10,-20)(-16,7)(1,9)(3,13)(5,11,19)
Loop annotated with half-edges
12^1_164 annotated with half-edges