$12^{2}_{334}$ - Minimal pinning sets
Pinning sets for 12^2_334 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_334 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 208
of which optimal: 2
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97933
on average over minimal pinning sets: 2.35556
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 4, 5}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 4, 8, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
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
2.2
6
0
1
13
2.55
7
0
0
40
2.79
8
0
0
61
2.97
9
0
0
54
3.1
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
1
205
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,6,3],[0,2,5,7],[0,8,8,1],[1,7,3,6],[1,5,2,2],[3,5,9,9],[4,9,9,4],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,13,18,14],[15,6,16,7],[19,5,20,6],[1,12,2,13],[18,9,19,8],[14,8,15,7],[9,4,10,5],[11,2,12,3],[3,10,4,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,16,-10,-1)(7,2,-8,-3)(18,3,-19,-4)(13,4,-14,-5)(5,12,-6,-13)(6,17,-7,-18)(1,8,-2,-9)(15,10,-16,-11)(20,11,-17,-12)(14,19,-15,-20)
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,-9)(-2,7,17,11,-16,9)(-3,18,-7)(-4,13,-6,-18)(-5,-13)(-8,1,-10,15,19,3)(-11,20,-15)(-12,5,-14,-20)(-17,6,12)(-19,14,4)(2,8)(10,16)
Multiloop annotated with half-edges
12^2_334 annotated with half-edges