$11^{2}_{53}$ - Minimal pinning sets
Pinning sets for 11^2_53 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_53 Pinning data
Pinning number of this multiloop: 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, 4, 8}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 4, 8, 9}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 3, 4, 7, 8}
5
[2, 2, 2, 2, 3]
2.20
D (optimal)
• {1, 3, 4, 5, 8}
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 multiloop 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],[0,5,5,2],[0,1,6,7],[0,7,8,8],[0,8,8,5],[1,4,6,1],[2,5,7,7],[2,6,6,3],[3,4,4,3]]
PD code (use to draw this multiloop with SnapPy): [[12,18,1,13],[13,7,14,8],[8,11,9,12],[17,3,18,4],[1,16,2,15],[6,14,7,15],[10,5,11,6],[9,5,10,4],[2,16,3,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(2,7,-3,-8)(8,3,-9,-4)(16,5,-17,-6)(6,9,-7,-10)(4,17,-5,-18)(18,11,-13,-12)(12,13,-1,-14)(10,15,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14)(-2,-8,-4,-18,-12,-14)(-3,8)(-5,16,-11,18)(-6,-10,-16)(-7,2,-15,10)(-9,6,-17,4)(-13,12)(1,13,11,15)(3,7,9)(5,17)
Multiloop annotated with half-edges
11^2_53 annotated with half-edges