$12^{1}_{172}$ - Minimal pinning sets
Pinning sets for 12^1_172 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_172 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 368
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04152
on average over minimal pinning sets: 2.4625
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)
• {1, 3, 5, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 4, 10, 11}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 3, 4, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
c (minimal)
• {1, 2, 3, 4, 11}
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
3
8
2.56
6
0
0
43
2.79
7
0
0
87
2.96
8
0
0
104
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
3
364
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,7],[0,8,6,5],[1,4,2,1],[2,4,9,3],[3,9,8,3],[4,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,18,11,17],[19,16,20,17],[3,8,4,9],[1,13,2,12],[18,12,19,11],[2,15,3,16],[7,4,8,5],[13,7,14,6],[14,5,15,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,20,-4,-1)(10,1,-11,-2)(2,9,-3,-10)(19,4,-20,-5)(16,5,-17,-6)(7,14,-8,-15)(8,17,-9,-18)(18,11,-19,-12)(15,12,-16,-13)(13,6,-14,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-3)(-2,-10)(-4,19,11,1)(-5,16,12,-19)(-6,13,-16)(-7,-15,-13)(-8,-18,-12,15)(-9,2,-11,18)(-14,7)(-17,8,14,6)(-20,3,9,17,5)(4,20)
Loop annotated with half-edges
12^1_172 annotated with half-edges