$12^{1}_{193}$ - Minimal pinning sets
Pinning sets for 12^1_193 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_193 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 184
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: 2.97614
on average over minimal pinning sets: 2.38333
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)
• {2, 3, 4, 6, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 5, 6, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 4, 6, 8, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
c (minimal)
• {1, 2, 4, 6, 7, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
3
7
2.5
7
0
0
33
2.76
8
0
0
54
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
1
3
180
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,6,7],[0,7,8,9],[0,9,9,5],[1,4,6,1],[1,5,2,2],[2,8,8,3],[3,7,7,9],[3,8,4,4]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,17,9,18],[19,14,20,15],[6,3,7,4],[1,11,2,10],[16,9,17,10],[18,16,19,15],[13,4,14,5],[5,12,6,13],[2,11,3,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (4,1,-5,-2)(11,2,-12,-3)(18,5,-19,-6)(14,7,-15,-8)(8,19,-9,-20)(20,9,-1,-10)(3,10,-4,-11)(17,12,-18,-13)(13,16,-14,-17)(6,15,-7,-16)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,10)(-2,11,-4)(-3,-11)(-5,18,12,2)(-6,-16,13,-18)(-7,14,16)(-8,-20,-10,3,-12,17,-14)(-9,20)(-13,-17)(-15,6,-19,8)(1,9,19,5)(7,15)
Loop annotated with half-edges
12^1_193 annotated with half-edges