$11^{3}_{9}$ - Minimal pinning sets
Pinning sets for 11^3_9 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^3_9 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, 3, 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, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,4,0],[0,4,4,5],[0,5,6,1],[1,7,2,2],[2,7,6,3],[3,5,8,8],[4,8,8,5],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[8,12,1,9],[9,7,10,8],[11,18,12,13],[1,6,2,7],[10,14,11,13],[5,17,6,18],[2,17,3,16],[14,4,15,5],[3,15,4,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,2,-14,-3)(7,4,-8,-5)(16,5,-17,-6)(6,15,-7,-16)(1,14,-2,-9)(9,8,-10,-1)(3,10,-4,-11)(18,11,-15,-12)(12,17,-13,-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,-9)(-2,13,17,5,-8,9)(-3,-11,18,-13)(-4,7,15,11)(-5,16,-7)(-6,-16)(-10,3,-14,1)(-12,-18)(-15,6,-17,12)(2,14)(4,10,8)
Multiloop annotated with half-edges
11^3_9 annotated with half-edges