$11^{2}_{68}$ - Minimal pinning sets
Pinning sets for 11^2_68 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_68 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 192
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04458
on average over minimal pinning sets: 2.56667
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 5, 6, 10}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 3, 5, 6, 11}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 2, 5, 6, 11}
5
[2, 2, 3, 3, 3]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.5
5
0
2
7
2.71
6
0
0
31
2.88
7
0
0
55
3.01
8
0
0
55
3.11
9
0
0
31
3.19
10
0
0
9
3.24
11
0
0
1
3.27
Total
1
2
189
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,7,7,8],[0,8,5,1],[1,4,8,2],[2,8,7,7],[3,6,6,3],[3,6,5,4]]
PD code (use to draw this multiloop with SnapPy): [[4,18,1,5],[5,13,6,12],[3,11,4,12],[17,8,18,9],[1,14,2,13],[6,2,7,3],[15,10,16,11],[9,16,10,17],[7,14,8,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(14,9,-15,-10)(1,10,-2,-11)(11,18,-12,-5)(12,3,-13,-4)(6,13,-7,-14)(8,15,-9,-16)(2,17,-3,-18)(16,7,-17,-8)
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,-11,-5)(-2,-18,11)(-3,12,18)(-4,5,-12)(-6,-14,-10,1)(-7,16,-9,14)(-8,-16)(-13,6,4)(-15,8,-17,2,10)(3,17,7,13)(9,15)
Multiloop annotated with half-edges
11^2_68 annotated with half-edges