$12^{2}_{290}$ - Minimal pinning sets
Pinning sets for 12^2_290 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_290 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03441
on average over minimal pinning sets: 2.46667
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, 3, 5, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 8, 10, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 3, 8, 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
1
0
0
2.4
6
0
2
7
2.63
7
0
0
31
2.83
8
0
0
55
2.99
9
0
0
55
3.12
10
0
0
31
3.22
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
2
189
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: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,4],[0,3,7,1],[1,7,6,2],[3,5,8,8],[4,9,9,5],[6,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[16,7,1,8],[8,17,9,20],[15,19,16,20],[6,18,7,19],[1,18,2,17],[9,14,10,15],[10,5,11,6],[2,13,3,14],[4,11,5,12],[12,3,13,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(8,3,-9,-4)(14,5,-15,-6)(2,9,-3,-10)(16,11,-1,-12)(7,12,-8,-13)(4,17,-5,-18)(13,18,-14,-19)(19,6,-20,-7)(20,15,-17,-16)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-3,8,12)(-2,-10)(-4,-18,13,-8)(-5,14,18)(-6,19,-14)(-7,-13,-19)(-9,2,-11,16,-17,4)(-12,7,-20,-16)(-15,20,6)(1,11)(3,9)(5,17,15)
Multiloop annotated with half-edges
12^2_290 annotated with half-edges