$11^{2}_{10}$ - Minimal pinning sets
Pinning sets for 11^2_10 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_10 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 146
of which optimal: 3
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9873
on average over minimal pinning sets: 2.59444
on average over optimal pinning sets: 2.46667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 7, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 6, 8, 11}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 3, 5, 8, 10, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
b (minimal)
• {1, 2, 3, 5, 8, 11}
6
[2, 2, 2, 3, 4, 4]
2.83
c (minimal)
• {1, 2, 3, 7, 8, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.47
6
0
3
17
2.72
7
0
0
43
2.92
8
0
0
45
3.06
9
0
0
26
3.15
10
0
0
8
3.23
11
0
0
1
3.27
Total
3
3
140
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,7],[0,7,8,8],[0,5,5,1],[1,4,4,6],[1,5,8,2],[2,8,3,2],[3,7,6,3]]
PD code (use to draw this multiloop with SnapPy): [[10,18,1,11],[11,14,12,15],[4,9,5,10],[17,7,18,8],[1,13,2,14],[12,2,13,3],[15,3,16,4],[8,5,9,6],[6,16,7,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(5,2,-6,-3)(13,4,-14,-5)(1,6,-2,-7)(16,9,-17,-10)(8,17,-9,-18)(15,18,-16,-11)(3,12,-4,-13)(7,14,-8,-15)
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,-7,-15,-11)(-2,5,-14,7)(-3,-13,-5)(-4,13)(-6,1,-12,3)(-8,-18,15)(-9,16,18)(-10,11,-16)(-17,8,14,4,12,10)(2,6)(9,17)
Multiloop annotated with half-edges
11^2_10 annotated with half-edges