$11^{2}_{69}$ - Minimal pinning sets
Pinning sets for 11^2_69 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_69 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)
• {1, 3, 6, 11}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 4, 6, 10}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 3, 4, 6, 10}
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, 3, 4, 5, 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],[0,6,6,7],[0,8,5,1],[1,4,8,2],[2,7,3,3],[3,6,8,8],[4,7,7,5]]
PD code (use to draw this multiloop with SnapPy): [[4,18,1,5],[5,14,6,15],[15,3,16,4],[10,17,11,18],[1,13,2,14],[6,2,7,3],[16,9,17,10],[11,9,12,8],[12,7,13,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(14,1,-15,-2)(7,10,-8,-11)(15,8,-16,-9)(3,12,-4,-13)(18,13,-5,-14)(9,16,-10,-17)(2,17,-3,-18)(11,6,-12,-7)
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,-5)(-2,-18,-14)(-3,-13,18)(-4,5,13)(-6,11,-8,15,1)(-7,-11)(-9,-17,2,-15)(-10,7,-12,3,17)(-16,9)(4,12,6)(8,10,16)
Multiloop annotated with half-edges
11^2_69 annotated with half-edges