$12^{1}_{190}$ - Minimal pinning sets
Pinning sets for 12^1_190 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_190 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, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 6, 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, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,7],[1,7,7,5],[2,4,8,8],[3,8,9,9],[3,9,4,4],[5,9,6,5],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[20,13,1,14],[14,19,15,20],[15,12,16,13],[1,16,2,17],[5,18,6,19],[6,11,7,12],[2,10,3,9],[17,4,18,5],[10,7,11,8],[3,8,4,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(7,4,-8,-5)(20,5,-1,-6)(6,19,-7,-20)(3,8,-4,-9)(12,9,-13,-10)(17,10,-18,-11)(18,13,-19,-14)(2,15,-3,-16)(11,16,-12,-17)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-19,6)(-2,-16,11,-18,-14)(-3,-9,12,16)(-4,7,19,13,9)(-5,20,-7)(-6,-20)(-8,3,15,1,5)(-10,17,-12)(-11,-17)(-13,18,10)(-15,2)(4,8)
Loop annotated with half-edges
12^1_190 annotated with half-edges