$11^{1}_{74}$ - Minimal pinning sets
Pinning sets for 11^1_74 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_74 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 100
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.92301
on average over minimal pinning sets: 2.44
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, 4, 8, 10}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 4, 5, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 3, 4, 6, 10}
6
[2, 2, 2, 2, 3, 4]
2.50
c (minimal)
• {1, 2, 4, 5, 9, 10}
6
[2, 2, 2, 2, 3, 4]
2.50
d (minimal)
• {1, 2, 4, 6, 9, 10}
6
[2, 2, 2, 2, 4, 4]
2.67
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
4
6
2.52
7
0
0
27
2.79
8
0
0
33
2.98
9
0
0
21
3.11
10
0
0
7
3.2
11
0
0
1
3.27
Total
1
4
95
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,5],[0,5,6,7],[1,7,7,8],[2,6,3,2],[3,5,8,8],[3,8,4,4],[4,7,6,6]]
PD code (use to draw this loop with SnapPy): [[11,18,12,1],[17,10,18,11],[12,10,13,9],[1,14,2,15],[5,16,6,17],[13,8,14,9],[2,8,3,7],[15,4,16,5],[6,4,7,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,3,-1,-4)(12,1,-13,-2)(4,17,-5,-18)(10,5,-11,-6)(6,9,-7,-10)(14,7,-15,-8)(16,11,-17,-12)(2,13,-3,-14)(8,15,-9,-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,-17,4)(-2,-14,-8,-16,-12)(-3,18,-5,10,-7,14)(-4,-18)(-6,-10)(-9,6,-11,16)(-13,2)(-15,8)(1,3,13)(5,17,11)(7,9,15)
Loop annotated with half-edges
11^1_74 annotated with half-edges