$11^{1}_{77}$ - Minimal pinning sets
Pinning sets for 11^1_77 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_77 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 88
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: 2.91292
on average over minimal pinning sets: 2.34444
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, 7, 8}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 6, 7, 8}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 4, 5, 7, 8}
6
[2, 2, 2, 2, 3, 4]
2.50
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
2
6
2.5
7
0
0
22
2.76
8
0
0
29
2.96
9
0
0
20
3.1
10
0
0
7
3.2
11
0
0
1
3.27
Total
1
2
85
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,5,6],[0,6,6,0],[1,7,5,1],[2,4,7,8],[2,8,3,3],[4,8,8,5],[5,7,7,6]]
PD code (use to draw this loop with SnapPy): [[7,18,8,1],[6,11,7,12],[17,10,18,11],[8,2,9,1],[12,5,13,6],[13,16,14,17],[9,2,10,3],[15,4,16,5],[14,4,15,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,11,-1,-12)(12,1,-13,-2)(9,2,-10,-3)(14,7,-15,-8)(3,8,-4,-9)(10,13,-11,-14)(4,15,-5,-16)(16,5,-17,-6)(6,17,-7,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,9,-4,-16,-6,-18,-12)(-3,-9)(-5,16)(-7,14,-11,18)(-8,3,-10,-14)(-13,10,2)(-15,4,8)(-17,6)(1,11,13)(5,15,7,17)
Loop annotated with half-edges
11^1_77 annotated with half-edges