$12^{2}_{303}$ - Minimal pinning sets
Pinning sets for 12^2_303 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_303 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 256
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04372
on average over minimal pinning sets: 2.475
on average over optimal pinning sets: 2.46667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 5, 6, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {2, 4, 5, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {2, 3, 5, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 5, 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
3
0
0
2.47
6
0
1
18
2.72
7
0
0
51
2.9
8
0
0
75
3.04
9
0
0
65
3.15
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
1
252
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: [[1,2,2,3],[0,3,4,5],[0,6,6,0],[0,7,4,1],[1,3,7,5],[1,4,8,6],[2,5,9,2],[3,9,8,4],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,14,8,15],[19,5,20,6],[1,13,2,14],[8,2,9,3],[15,3,16,4],[4,18,5,19],[12,9,13,10],[16,12,17,11],[17,10,18,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(14,1,-15,-2)(18,3,-19,-4)(16,9,-17,-10)(10,15,-11,-16)(5,12,-6,-13)(20,13,-7,-14)(2,17,-3,-18)(4,19,-5,-20)(11,8,-12,-9)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-7)(-2,-18,-4,-20,-14)(-3,18)(-5,-13,20)(-6,7,13)(-8,11,15,1)(-9,16,-11)(-10,-16)(-12,5,19,3,17,9)(-15,10,-17,2)(-19,4)(6,12,8)
Multiloop annotated with half-edges
12^2_303 annotated with half-edges