$11^{1}_{18}$ - Minimal pinning sets
Pinning sets for 11^1_18 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_18 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 120
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91164
on average over minimal pinning sets: 2.3
on average over optimal pinning sets: 2.3
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 7, 10}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 5, 10}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 3, 4, 10}
5
[2, 2, 2, 2, 3]
2.20
D (optimal)
• {1, 2, 3, 6, 10}
5
[2, 2, 2, 2, 4]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.3
6
0
0
18
2.63
7
0
0
34
2.85
8
0
0
35
3.0
9
0
0
21
3.11
10
0
0
7
3.2
11
0
0
1
3.27
Total
4
0
116
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,5,5,3],[0,2,6,7],[0,7,6,1],[1,2,2,1],[3,4,8,8],[3,8,8,4],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[9,18,10,1],[15,8,16,9],[17,6,18,7],[10,6,11,5],[1,14,2,15],[7,16,8,17],[11,2,12,3],[13,4,14,5],[12,4,13,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,9,-1,-10)(10,1,-11,-2)(2,7,-3,-8)(14,3,-15,-4)(12,5,-13,-6)(16,11,-17,-12)(4,13,-5,-14)(6,15,-7,-16)(8,17,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,-8,-18,-10)(-3,14,-5,12,-17,8)(-4,-14)(-6,-16,-12)(-7,2,-11,16)(-9,18)(-13,4,-15,6)(1,9,17,11)(3,7,15)(5,13)
Loop annotated with half-edges
11^1_18 annotated with half-edges