$12^{2}_{249}$ - Minimal pinning sets
Pinning sets for 12^2_249 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_249 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 92
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91906
on average over minimal pinning sets: 2.32738
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 4, 5, 7, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
a (minimal)
• {1, 2, 4, 5, 6, 7, 11}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
b (minimal)
• {1, 2, 4, 5, 7, 10, 11}
7
[2, 2, 2, 2, 2, 3, 4]
2.43
c (minimal)
• {1, 2, 4, 5, 7, 9, 11}
7
[2, 2, 2, 2, 2, 3, 4]
2.43
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
1
0
0
2.17
7
0
3
6
2.46
8
0
0
24
2.76
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
3
88
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,5,5],[0,6,6,0],[1,7,5,1],[2,4,8,2],[3,9,7,3],[4,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[9,16,10,1],[8,20,9,17],[15,19,16,20],[10,2,11,1],[17,7,18,8],[18,14,19,15],[2,12,3,11],[3,6,4,7],[4,13,5,14],[12,5,13,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,3,-11,-4)(1,4,-2,-5)(18,5,-19,-6)(9,16,-10,-1)(2,11,-3,-12)(19,12,-20,-13)(7,14,-8,-15)(15,8,-16,-9)(13,20,-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)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-5,18,-7,-15,-9)(-2,-12,19,5)(-3,10,16,8,14,20,12)(-4,1,-10)(-6,-18)(-8,15)(-11,2,4)(-13,-17,6,-19)(-14,7,17)(-16,9)(-20,13)(3,11)
Multiloop annotated with half-edges
12^2_249 annotated with half-edges