$12^{2}_{335}$ - Minimal pinning sets
Pinning sets for 12^2_335 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_335 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 240
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97778
on average over minimal pinning sets: 2.3
on average over optimal pinning sets: 2.3
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 6, 8, 9}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 5, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
D (optimal)
• {1, 2, 4, 8, 9}
5
[2, 2, 2, 2, 4]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.3
6
0
0
22
2.62
7
0
0
52
2.84
8
0
0
69
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
236
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,5,6,0],[0,6,7,1],[1,7,7,5],[1,4,8,2],[2,9,9,3],[3,8,4,4],[5,7,9,9],[6,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[5,14,6,1],[4,20,5,15],[13,6,14,7],[1,16,2,15],[3,8,4,9],[19,7,20,8],[12,16,13,17],[2,10,3,9],[10,18,11,19],[17,11,18,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,4,-12,-5)(20,5,-15,-6)(1,6,-2,-7)(7,14,-8,-1)(18,9,-19,-10)(3,12,-4,-13)(10,17,-11,-18)(8,19,-9,-20)(2,15,-3,-16)(16,13,-17,-14)
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)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-7)(-2,-16,-14,7)(-3,-13,16)(-4,11,17,13)(-5,20,-9,18,-11)(-6,1,-8,-20)(-10,-18)(-12,3,15,5)(-15,2,6)(-17,10,-19,8,14)(4,12)(9,19)
Multiloop annotated with half-edges
12^2_335 annotated with half-edges