$12^{2}_{190}$ - Minimal pinning sets
Pinning sets for 12^2_190 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_190 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, 6, 9, 12}
5
[2, 2, 2, 2, 4]
2.40
B (optimal)
• {1, 2, 6, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
C (optimal)
• {1, 2, 6, 9, 11}
5
[2, 2, 2, 2, 4]
2.40
D (optimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
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: [[1,2,3,4],[0,5,5,2],[0,1,6,7],[0,7,8,8],[0,8,9,6],[1,6,6,1],[2,5,5,4],[2,9,9,3],[3,9,4,3],[4,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[16,11,1,12],[12,6,13,5],[15,4,16,5],[10,20,11,17],[1,9,2,8],[6,14,7,13],[7,14,8,15],[3,17,4,18],[19,9,20,10],[2,19,3,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,2,-14,-3)(3,10,-4,-11)(6,15,-7,-16)(18,7,-19,-8)(14,9,-15,-10)(11,4,-12,-5)(5,12,-6,-13)(17,16,-18,-1)(8,19,-9,-20)(1,20,-2,-17)
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,-17)(-2,13,-6,-16,17)(-3,-11,-5,-13)(-4,11)(-7,18,16)(-8,-20,1,-18)(-9,14,2,20)(-10,3,-14)(-12,5)(-15,6,12,4,10)(-19,8)(7,15,9,19)
Multiloop annotated with half-edges
12^2_190 annotated with half-edges