$12^{1}_{229}$ - Minimal pinning sets
Pinning sets for 12^1_229 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_229 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, 8, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 7, 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],[0,5,6,2],[0,1,3,3],[0,2,2,7],[0,8,5,5],[1,4,4,8],[1,8,9,9],[3,9,9,8],[4,7,6,5],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[17,20,18,1],[3,16,4,17],[4,19,5,20],[18,5,19,6],[1,10,2,11],[11,2,12,3],[15,8,16,9],[6,14,7,13],[9,12,10,13],[7,14,8,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(16,3,-17,-4)(13,6,-14,-7)(19,8,-20,-9)(7,10,-8,-11)(11,4,-12,-5)(5,12,-6,-13)(1,14,-2,-15)(2,17,-3,-18)(15,18,-16,-19)
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,-15,-19,-9)(-2,-18,15)(-3,16,18)(-4,11,-8,19,-16)(-5,-13,-7,-11)(-6,13)(-10,7,-14,1)(-12,5)(-17,2,14,6,12,4)(-20,9)(3,17)(8,10,20)
Loop annotated with half-edges
12^1_229 annotated with half-edges