$11^{3}_{1}$ - Minimal pinning sets
Pinning sets for 11^3_1 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^3_1 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 64
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: 2.83846
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, 10}
5
[2, 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
5
1
0
0
2.0
6
0
0
6
2.39
7
0
0
15
2.67
8
0
0
20
2.88
9
0
0
15
3.04
10
0
0
6
3.17
11
0
0
1
3.27
Total
1
0
63
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,6,6,7],[0,7,4,4],[1,3,3,7],[1,7,8,8],[2,8,8,2],[2,5,4,3],[5,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[4,10,1,5],[5,3,6,4],[9,18,10,11],[1,15,2,16],[16,2,17,3],[6,14,7,13],[11,8,12,9],[14,17,15,18],[7,12,8,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(14,1,-15,-2)(8,17,-9,-18)(2,13,-3,-14)(10,15,-11,-16)(3,12,-4,-5)(11,6,-12,-7)(18,7,-13,-8)(16,9,-17,-10)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-3,-5)(-2,-14)(-4,5)(-6,11,15,1)(-7,18,-9,16,-11)(-8,-18)(-10,-16)(-12,3,13,7)(-13,2,-15,10,-17,8)(4,12,6)(9,17)
Multiloop annotated with half-edges
11^3_1 annotated with half-edges