$11^{1}_{21}$ - Minimal pinning sets
Pinning sets for 11^1_21 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_21 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: 2.92668
on average over minimal pinning sets: 2.34667
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 6, 10}
5
[2, 2, 2, 2, 4]
2.40
a (minimal)
• {1, 2, 3, 8, 10, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 3, 7, 8, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
c (minimal)
• {1, 2, 3, 4, 10, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
d (minimal)
• {1, 2, 3, 4, 7, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
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
2.4
6
0
4
6
2.53
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, 3, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,7,7],[0,5,1,1],[1,4,6,2],[2,5,8,8],[3,8,8,3],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[3,18,4,1],[2,9,3,10],[12,17,13,18],[4,13,5,14],[1,11,2,10],[11,8,12,9],[16,7,17,8],[5,15,6,14],[6,15,7,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,5,-15,-6)(9,6,-10,-7)(18,7,-1,-8)(8,17,-9,-18)(2,11,-3,-12)(12,3,-13,-4)(4,13,-5,-14)(10,15,-11,-16)(1,16,-2,-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,-17,8)(-2,-12,-4,-14,-6,9,17)(-3,12)(-5,14)(-7,18,-9)(-8,-18)(-10,-16,1,7)(-11,2,16)(-13,4)(-15,10,6)(3,11,15,5,13)
Loop annotated with half-edges
11^1_21 annotated with half-edges