$11^{3}_{6}$ - Minimal pinning sets
Pinning sets for 11^3_6 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^3_6 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 128
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.89692
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 7, 8}
4
[2, 2, 2, 2]
2.00
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
2.0
5
0
0
7
2.4
6
0
0
21
2.67
7
0
0
35
2.86
8
0
0
35
3.0
9
0
0
21
3.11
10
0
0
7
3.2
11
0
0
1
3.27
Total
1
0
127
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,4,0],[0,4,5,6],[0,6,7,1],[1,8,8,2],[2,8,8,7],[2,7,7,3],[3,6,6,5],[4,5,5,4]]
PD code (use to draw this multiloop with SnapPy): [[6,12,1,7],[7,5,8,6],[11,18,12,13],[1,4,2,5],[8,14,9,13],[15,10,16,11],[3,17,4,18],[2,17,3,16],[14,10,15,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,2,-6,-3)(14,3,-15,-4)(12,17,-7,-18)(4,13,-5,-14)(16,11,-17,-12)(7,6,-8,-1)(1,8,-2,-9)(18,9,-13,-10)(10,15,-11,-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,-9,18,-7)(-2,5,13,9)(-3,14,-5)(-4,-14)(-6,7,17,11,15,3)(-8,1)(-10,-16,-12,-18)(-11,16)(-13,4,-15,10)(-17,12)(2,8,6)
Multiloop annotated with half-edges
11^3_6 annotated with half-edges