$12^{2}_{207}$ - Minimal pinning sets
Pinning sets for 12^2_207 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_207 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 240
of which optimal: 2
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03455
on average over minimal pinning sets: 2.45
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 5, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 3, 5, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.4
6
0
2
13
2.66
7
0
0
45
2.86
8
0
0
71
3.02
9
0
0
64
3.14
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
2
236
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,4],[1,3,6,7],[2,7,8,8],[3,9,7,4],[4,6,9,5],[5,9,9,5],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,9,12,10],[12,19,13,20],[1,13,2,14],[14,8,15,9],[4,18,5,19],[2,7,3,8],[15,3,16,4],[17,5,18,6],[6,16,7,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-19,-2)(20,3,-11,-4)(10,5,-1,-6)(17,6,-18,-7)(14,7,-15,-8)(9,16,-10,-17)(4,19,-5,-20)(2,11,-3,-12)(15,12,-16,-13)(8,13,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,6)(-2,-12,15,7,-18)(-3,20,-5,10,16,12)(-4,-20)(-6,17,-10)(-7,14,-9,-17)(-8,-14)(-11,2,-19,4)(-13,8,-15)(-16,9,13)(1,5,19)(3,11)
Multiloop annotated with half-edges
12^2_207 annotated with half-edges