$11^{2}_{33}$ - Minimal pinning sets
Pinning sets for 11^2_33 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_33 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 48
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: 2.84761
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 7, 10}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 4, 6, 10}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
9
2.54
8
0
0
16
2.81
9
0
0
14
3.02
10
0
0
6
3.17
11
0
0
1
3.27
Total
2
0
46
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,5,6],[0,6,6,0],[0,5,7,7],[1,8,8,5],[1,4,3,6],[1,5,2,2],[3,8,8,3],[4,7,7,4]]
PD code (use to draw this multiloop with SnapPy): [[8,18,1,9],[9,5,10,6],[17,7,18,8],[1,11,2,12],[14,4,15,5],[10,15,11,16],[6,16,7,17],[2,13,3,12],[3,13,4,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(15,2,-16,-3)(13,4,-14,-5)(18,7,-9,-8)(8,9,-1,-10)(16,11,-17,-12)(3,12,-4,-13)(5,14,-6,-15)(6,17,-7,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,15,-6,-18,-8,-10)(-3,-13,-5,-15)(-4,13)(-7,18)(-9,8)(-11,16,2)(-12,3,-16)(-14,5)(-17,6,14,4,12)(1,9,7,17,11)
Multiloop annotated with half-edges
11^2_33 annotated with half-edges