$11^{1}_{41}$ - Minimal pinning sets
Pinning sets for 11^1_41 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_41 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 176
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97775
on average over minimal pinning sets: 2.41667
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 6}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 3, 6, 10}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 3, 4, 6, 10}
5
[2, 2, 2, 3, 4]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
2
7
2.56
6
0
0
30
2.78
7
0
0
51
2.95
8
0
0
49
3.07
9
0
0
27
3.16
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
2
173
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,4,0],[0,4,5,6],[0,6,7,1],[1,5,5,2],[2,4,4,8],[2,8,7,3],[3,6,8,8],[5,7,7,6]]
PD code (use to draw this loop with SnapPy): [[7,18,8,1],[17,6,18,7],[8,11,9,12],[1,16,2,17],[10,5,11,6],[9,5,10,4],[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)(2,7,-3,-8)(8,3,-9,-4)(15,4,-16,-5)(6,9,-7,-10)(18,11,-1,-12)(10,13,-11,-14)(17,14,-18,-15)(5,16,-6,-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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,-8,-4,15,-18,-12)(-3,8)(-5,-17,-15)(-6,-10,-14,17)(-7,2,-13,10)(-9,6,16,4)(-11,18,14)(-16,5)(1,11,13)(3,7,9)
Loop annotated with half-edges
11^1_41 annotated with half-edges