$12^{2}_{254}$ - Minimal pinning sets
Pinning sets for 12^2_254 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_254 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 160
of which optimal: 5
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.985
on average over minimal pinning sets: 2.4
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, 3, 4, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 7, 11, 12}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 3, 7, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
D (optimal)
• {1, 2, 3, 7, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
E (optimal)
• {1, 2, 3, 5, 8, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
5
0
0
2.4
7
0
0
24
2.7
8
0
0
47
2.92
9
0
0
48
3.08
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
5
0
155
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,5],[0,6,6,3],[0,2,7,0],[1,8,8,5],[1,4,9,1],[2,9,7,2],[3,6,9,8],[4,7,9,4],[5,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,7,16,8],[13,2,14,3],[19,1,20,2],[6,9,7,10],[16,9,17,8],[3,12,4,13],[4,18,5,19],[10,5,11,6],[17,11,18,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,2,-8,-3)(15,4,-16,-5)(5,16,-6,-17)(3,6,-4,-7)(17,8,-18,-9)(14,9,-1,-10)(19,12,-20,-13)(10,13,-11,-14)(11,20,-12,-15)(1,18,-2,-19)
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,-19,-13,10)(-2,7,-4,15,-12,19)(-3,-7)(-5,-17,-9,14,-11,-15)(-6,3,-8,17)(-10,-14)(-16,5)(-18,1,9)(-20,11,13)(2,18,8)(4,6,16)(12,20)
Multiloop annotated with half-edges
12^2_254 annotated with half-edges