$12^{1}_{191}$ - Minimal pinning sets
Pinning sets for 12^1_191 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_191 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)
• {1, 2, 7, 8, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 6, 7, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
b (minimal)
• {1, 2, 4, 7, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
c (minimal)
• {1, 2, 3, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
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, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,6,7],[0,7,7,8],[0,8,9,9],[1,9,8,1],[1,7,2,2],[2,6,3,3],[3,5,9,4],[4,8,5,4]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[17,4,18,5],[19,8,20,9],[6,11,7,12],[1,15,2,14],[3,16,4,17],[18,10,19,9],[10,7,11,8],[12,16,13,15],[2,13,3,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,20,-14,-1)(6,3,-7,-4)(15,4,-16,-5)(5,14,-6,-15)(2,7,-3,-8)(18,9,-19,-10)(10,17,-11,-18)(11,8,-12,-9)(19,12,-20,-13)(1,16,-2,-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,-17,10,-19,-13)(-2,-8,11,17)(-3,6,14,20,12,8)(-4,15,-6)(-5,-15)(-7,2,16,4)(-9,18,-11)(-10,-18)(-12,19,9)(-14,5,-16,1)(-20,13)(3,7)
Loop annotated with half-edges
12^1_191 annotated with half-edges