$12^{3}_{71}$ - Minimal pinning sets
Pinning sets for 12^3_71 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_71 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, 4, 5, 9}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 7, 10, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 4, 7, 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, 5, 5, 5]
Minimal region degree: 2
Is multisimple: Yes
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],[1,8,8,2],[3,8,9,9],[4,9,9,8],[5,7,6,5],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,17,10,20],[7,19,8,20],[15,18,16,19],[1,18,2,17],[10,6,11,7],[14,4,15,5],[2,13,3,12],[5,11,6,12],[3,13,4,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(13,2,-14,-3)(1,14,-2,-15)(15,6,-16,-7)(7,16,-8,-9)(4,11,-5,-12)(10,17,-11,-18)(3,18,-4,-19)(19,12,-20,-13)(20,5,-17,-6)
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,-15,-7,-9)(-2,13,-20,-6,15)(-3,-19,-13)(-4,-12,19)(-5,20,12)(-8,9)(-10,-18,3,-14,1)(-11,4,18)(-16,7)(-17,10,8,16,6)(2,14)(5,11,17)
Multiloop annotated with half-edges
12^3_71 annotated with half-edges