$11^{1}_{84}$ - Minimal pinning sets
Pinning sets for 11^1_84 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_84 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 56
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.85514
on average over minimal pinning sets: 2.22222
on average over optimal pinning sets: 2.22222
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 5, 7, 8}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 3, 4, 5, 8, 10}
6
[2, 2, 2, 2, 2, 3]
2.17
C (optimal)
• {1, 3, 4, 5, 8, 9}
6
[2, 2, 2, 2, 2, 4]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.22
7
0
0
12
2.6
8
0
0
19
2.86
9
0
0
15
3.04
10
0
0
6
3.17
11
0
0
1
3.27
Total
3
0
53
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,5,6,3],[0,2,7,7],[1,5,5,1],[2,4,4,8],[2,8,8,7],[3,6,8,3],[5,7,6,6]]
PD code (use to draw this loop with SnapPy): [[11,18,12,1],[17,10,18,11],[12,8,13,7],[1,7,2,6],[9,16,10,17],[8,16,9,15],[13,4,14,5],[2,5,3,6],[3,14,4,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(17,2,-18,-3)(3,16,-4,-17)(4,9,-5,-10)(14,7,-15,-8)(18,11,-1,-12)(12,5,-13,-6)(6,13,-7,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-5,12)(-2,17,-4,-10)(-3,-17)(-6,-14,-8,-16,3,-18,-12)(-7,14)(-9,4,16)(-11,18,2)(-13,6)(-15,8)(1,11)(5,9,15,7,13)
Loop annotated with half-edges
11^1_84 annotated with half-edges