$12^{2}_{97}$ - Minimal pinning sets
Pinning sets for 12^2_97 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_97 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 448
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04602
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 9}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 2, 6, 9}
4
[2, 2, 2, 3]
2.25
C (optimal)
• {1, 2, 5, 9}
4
[2, 2, 2, 4]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
3
0
0
2.33
5
0
0
21
2.65
6
0
0
64
2.85
7
0
0
111
3.0
8
0
0
120
3.1
9
0
0
83
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
0
445
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,5,6,1],[2,6,7,7],[2,8,6,3],[3,5,9,4],[4,9,8,4],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,7,10,8],[10,19,11,20],[1,6,2,7],[13,18,14,19],[11,5,12,6],[2,12,3,13],[17,14,18,15],[4,16,5,17],[3,16,4,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(5,2,-6,-3)(14,3,-15,-4)(6,17,-7,-18)(10,19,-11,-20)(20,7,-9,-8)(8,9,-1,-10)(18,11,-19,-12)(15,12,-16,-13)(4,13,-5,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,12,-19,10)(-2,5,13,-16)(-3,14,-5)(-4,-14)(-6,-18,-12,15,3)(-7,20,-11,18)(-8,-10,-20)(-9,8)(-13,4,-15)(-17,6,2)(1,9,7,17)(11,19)
Multiloop annotated with half-edges
12^2_97 annotated with half-edges