$11^{3}_{15}$ - Minimal pinning sets
Pinning sets for 11^3_15 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^3_15 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)
• {2, 4, 6, 10}
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, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,5,3],[0,2,6,7],[1,7,7,8],[1,8,2,2],[3,8,8,7],[3,6,4,4],[4,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[10,14,1,11],[11,9,12,10],[5,13,6,14],[1,6,2,7],[8,18,9,15],[12,4,13,5],[2,17,3,16],[7,16,8,15],[3,17,4,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(9,4,-10,-5)(18,7,-15,-8)(3,16,-4,-17)(8,17,-9,-18)(1,12,-2,-13)(6,13,-7,-14)(14,5,-11,-6)(11,10,-12,-1)
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,-13,6,-11)(-2,15,7,13)(-3,-17,8,-15)(-4,9,17)(-5,14,-7,18,-9)(-6,-14)(-8,-18)(-10,11,5)(-12,1)(-16,3)(2,12,10,4,16)
Multiloop annotated with half-edges
11^3_15 annotated with half-edges