$12^{1}_{199}$ - Minimal pinning sets
Pinning sets for 12^1_199 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_199 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 192
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: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 5, 11}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,7],[0,7,8,4],[0,3,8,6],[1,6,6,1],[2,5,5,4],[2,9,9,3],[3,9,9,4],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,9,17,10],[10,19,11,20],[5,14,6,15],[1,6,2,7],[8,17,9,18],[18,7,19,8],[11,4,12,5],[13,2,14,3],[3,12,4,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(9,2,-10,-3)(19,4,-20,-5)(16,5,-17,-6)(14,7,-15,-8)(1,10,-2,-11)(3,12,-4,-13)(18,13,-19,-14)(6,15,-7,-16)(8,17,-9,-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)(-2,9,17,5,-20,11)(-3,-13,18,-9)(-4,19,13)(-5,16,-7,14,-19)(-6,-16)(-8,-18,-14)(-10,1,-12,3)(-15,6,-17,8)(2,10)(4,12,20)(7,15)
Loop annotated with half-edges
12^1_199 annotated with half-edges