$11^{2}_{12}$ - Minimal pinning sets
Pinning sets for 11^2_12 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_12 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 160
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: 2.97092
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 4, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
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
2.25
5
0
1
7
2.55
6
0
0
26
2.77
7
0
0
45
2.93
8
0
0
45
3.06
9
0
0
26
3.15
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
1
158
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,3],[0,2,7,8],[0,6,5,5],[1,4,4,1],[1,4,8,2],[2,8,8,3],[3,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[8,18,1,9],[9,3,10,4],[13,7,14,8],[14,17,15,18],[1,12,2,11],[2,10,3,11],[4,12,5,13],[16,6,17,7],[15,6,16,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-9,-2)(3,14,-4,-15)(15,4,-16,-5)(12,5,-13,-6)(8,9,-1,-10)(10,7,-11,-8)(2,11,-3,-12)(13,16,-14,-17)(6,17,-7,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-7,10)(-2,-12,-6,-18)(-3,-15,-5,12)(-4,15)(-8,-10)(-9,8,-11,2)(-13,-17,6)(-14,3,11,7,17)(-16,13,5)(1,9)(4,14,16)
Multiloop annotated with half-edges
11^2_12 annotated with half-edges