$12^{2}_{99}$ - Minimal pinning sets
Pinning sets for 12^2_99 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_99 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)
• {2, 4, 5, 8, 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, 3, 3, 6, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,3],[0,2,7,7],[0,5,1,1],[1,4,8,9],[2,9,9,2],[3,8,8,3],[5,7,7,9],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,20,3,11],[15,9,16,10],[4,16,5,17],[1,12,2,11],[12,19,13,20],[8,14,9,15],[5,18,6,17],[6,18,7,19],[13,7,14,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,5,-15,-6)(16,7,-17,-8)(1,8,-2,-9)(6,15,-7,-16)(2,17,-3,-18)(18,3,-19,-4)(4,19,-5,-20)(13,20,-14,-11)(10,11,-1,-12)(12,9,-13,-10)
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,-9,12)(-2,-18,-4,-20,13,9)(-3,18)(-5,14,20)(-6,-16,-8,1,11,-14)(-7,16)(-10,-12)(-11,10,-13)(-15,6)(-17,2,8)(-19,4)(3,17,7,15,5,19)
Multiloop annotated with half-edges
12^2_99 annotated with half-edges