$11^{2}_{23}$ - Minimal pinning sets
Pinning sets for 11^2_23 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_23 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96934
on average over minimal pinning sets: 2.25
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 8}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 2, 5, 8}
4
[2, 2, 2, 3]
2.25
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.25
5
0
0
13
2.58
6
0
0
36
2.81
7
0
0
55
2.96
8
0
0
50
3.08
9
0
0
27
3.16
10
0
0
8
3.23
11
0
0
1
3.27
Total
2
0
190
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,3],[0,2,6,7],[0,5,1,1],[1,4,7,8],[2,8,3,2],[3,8,8,5],[5,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[3,12,4,1],[2,18,3,13],[6,11,7,12],[4,7,5,8],[1,14,2,13],[14,17,15,18],[10,5,11,6],[8,16,9,17],[15,9,16,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,3,-9,-4)(2,5,-3,-6)(16,7,-17,-8)(4,9,-5,-10)(1,10,-2,-11)(6,17,-7,-18)(15,18,-16,-13)(12,13,-1,-14)(14,11,-15,-12)
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,-11,14)(-2,-6,-18,15,11)(-3,8,-17,6)(-4,-10,1,13,-16,-8)(-5,2,10)(-7,16,18)(-9,4)(-12,-14)(-13,12,-15)(3,5,9)(7,17)
Multiloop annotated with half-edges
11^2_23 annotated with half-edges