$11^{1}_{27}$ - Minimal pinning sets
Pinning sets for 11^1_27 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_27 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, 2, 6, 10}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 3, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 2, 4, 9, 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, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,5,6,1],[2,6,7,7],[2,8,6,3],[3,5,8,4],[4,8,8,4],[5,7,7,6]]
PD code (use to draw this loop with SnapPy): [[18,7,1,8],[8,17,9,18],[9,6,10,7],[1,16,2,17],[12,5,13,6],[10,15,11,16],[2,11,3,12],[4,13,5,14],[14,3,15,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(5,2,-6,-3)(14,3,-15,-4)(16,7,-17,-8)(18,9,-1,-10)(6,11,-7,-12)(15,12,-16,-13)(4,13,-5,-14)(10,17,-11,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-17,10)(-2,5,13,-16,-8)(-3,14,-5)(-4,-14)(-6,-12,15,3)(-7,16,12)(-9,18,-11,6,2)(-10,-18)(-13,4,-15)(1,9)(7,11,17)
Loop annotated with half-edges
11^1_27 annotated with half-edges