$12^{2}_{229}$ - Minimal pinning sets
Pinning sets for 12^2_229 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_229 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 128
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90623
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 6, 9}
5
[2, 2, 2, 2, 2]
2.00
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
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,7,7,8],[0,8,8,4],[0,3,9,9],[1,9,6,1],[1,5,7,7],[2,6,6,2],[2,9,3,3],[4,8,5,4]]
PD code (use to draw this multiloop with SnapPy): [[5,16,6,1],[4,11,5,12],[15,8,16,9],[6,17,7,20],[1,20,2,19],[12,3,13,4],[13,10,14,11],[9,14,10,15],[7,17,8,18],[2,18,3,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,16,-10,-1)(12,5,-13,-6)(10,7,-11,-8)(1,8,-2,-9)(6,11,-7,-12)(2,13,-3,-14)(15,18,-16,-19)(19,14,-20,-15)(20,3,-17,-4)(4,17,-5,-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,-9)(-2,-14,19,-16,9)(-3,20,14)(-4,-18,15,-20)(-5,12,-7,10,16,18)(-6,-12)(-8,1,-10)(-11,6,-13,2,8)(-15,-19)(-17,4)(3,13,5,17)(7,11)
Multiloop annotated with half-edges
12^2_229 annotated with half-edges