$11^{2}_{32}$ - Minimal pinning sets
Pinning sets for 11^2_32 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_32 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 40
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.85151
on average over minimal pinning sets: 2.22619
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, 8}
6
[2, 2, 2, 2, 2, 3]
2.17
a (minimal)
• {1, 2, 3, 4, 5, 8, 9}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
1
0
0
2.17
7
0
1
5
2.48
8
0
0
13
2.76
9
0
0
13
2.99
10
0
0
6
3.17
11
0
0
1
3.27
Total
1
1
38
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,3],[0,2,6,6],[0,7,7,5],[1,4,2,1],[3,8,8,3],[4,8,8,4],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[10,18,1,11],[11,8,12,7],[9,6,10,7],[17,5,18,6],[1,14,2,13],[8,13,9,12],[4,16,5,17],[14,3,15,2],[15,3,16,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,11,-1,-12)(12,1,-13,-2)(14,3,-15,-4)(7,4,-8,-5)(5,16,-6,-17)(17,6,-18,-7)(18,9,-11,-10)(2,13,-3,-14)(8,15,-9,-16)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,-14,-4,7,-18,-10,-12)(-3,14)(-5,-17,-7)(-6,17)(-8,-16,5)(-9,18,6,16)(-11,10)(-13,2)(-15,8,4)(1,11,9,15,3,13)
Multiloop annotated with half-edges
11^2_32 annotated with half-edges