$11^{1}_{42}$ - Minimal pinning sets
Pinning sets for 11^1_42 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_42 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 96
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: 2.90403
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, 3, 6, 10}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 5, 6}
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
11
2.55
7
0
0
25
2.79
8
0
0
30
2.97
9
0
0
20
3.1
10
0
0
7
3.2
11
0
0
1
3.27
Total
2
0
94
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,4,0],[0,5,5,6],[0,6,7,1],[1,8,5,5],[2,4,4,2],[2,8,7,3],[3,6,8,8],[4,7,7,6]]
PD code (use to draw this loop with SnapPy): [[18,7,1,8],[8,17,9,18],[11,6,12,7],[1,16,2,17],[9,4,10,5],[5,10,6,11],[12,15,13,16],[2,13,3,14],[14,3,15,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(7,2,-8,-3)(9,4,-10,-5)(16,5,-17,-6)(3,8,-4,-9)(18,11,-1,-12)(10,13,-11,-14)(17,14,-18,-15)(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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,7,15,-18,-12)(-3,-9,-5,16,-7)(-4,9)(-6,-16)(-8,3)(-10,-14,17,5)(-11,18,14)(-13,10,4,8,2)(-15,6,-17)(1,11,13)
Loop annotated with half-edges
11^1_42 annotated with half-edges