$11^{2}_{3}$ - Minimal pinning sets
Pinning sets for 11^2_3 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_3 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 240
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: 2.97978
on average over minimal pinning sets: 2.375
on average over optimal pinning sets: 2.375
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 6, 8}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {2, 4, 7, 8}
4
[2, 2, 2, 4]
2.50
C (optimal)
• {2, 4, 8, 11}
4
[2, 2, 2, 3]
2.25
D (optimal)
• {2, 4, 8, 10}
4
[2, 2, 2, 4]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
4
0
0
2.38
5
0
0
22
2.67
6
0
0
52
2.87
7
0
0
69
3.0
8
0
0
56
3.09
9
0
0
28
3.17
10
0
0
8
3.23
11
0
0
1
3.27
Total
4
0
236
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,7],[0,7,8,8],[0,5,1,1],[1,4,8,6],[2,5,7,2],[2,6,8,3],[3,7,5,3]]
PD code (use to draw this multiloop with SnapPy): [[3,12,4,1],[2,7,3,8],[11,18,12,13],[4,15,5,16],[1,9,2,8],[9,6,10,7],[13,10,14,11],[14,17,15,18],[5,17,6,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,4,-8,-5)(12,5,-1,-6)(6,11,-7,-12)(17,8,-18,-9)(1,10,-2,-11)(2,13,-3,-14)(14,3,-15,-4)(18,15,-13,-16)(9,16,-10,-17)
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,-11,6)(-2,-14,-4,7,11)(-3,14)(-5,12,-7)(-6,-12)(-8,17,-10,1,5)(-9,-17)(-13,2,10,16)(-15,18,8,4)(-16,9,-18)(3,13,15)
Multiloop annotated with half-edges
11^2_3 annotated with half-edges