$12^{2}_{259}$ - Minimal pinning sets
Pinning sets for 12^2_259 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_259 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 124
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: 2.91549
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 6, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 6, 9, 11}
6
[2, 2, 2, 2, 2, 4]
2.33
C (optimal)
• {1, 2, 3, 6, 9, 10}
6
[2, 2, 2, 2, 2, 4]
2.33
D (optimal)
• {1, 2, 3, 6, 7, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
E (optimal)
• {1, 2, 3, 6, 8, 9}
6
[2, 2, 2, 2, 2, 4]
2.33
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
2.27
7
0
0
20
2.63
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
5
0
119
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,5,5,3],[0,2,6,0],[1,7,8,1],[1,8,2,2],[3,9,7,7],[4,6,6,9],[4,9,9,5],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,11,16,12],[13,2,14,3],[19,1,20,2],[10,16,11,17],[12,4,13,3],[6,18,7,19],[17,7,18,8],[9,4,10,5],[5,8,6,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,14,-12,-1)(1,10,-2,-11)(7,2,-8,-3)(20,5,-15,-6)(3,6,-4,-7)(18,9,-19,-10)(16,13,-17,-14)(8,19,-9,-20)(4,15,-5,-16)(12,17,-13,-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,-11)(-2,7,-4,-16,-14,11)(-3,-7)(-5,20,-9,18,-13,16)(-6,3,-8,-20)(-10,1,-12,-18)(-15,4,6)(-17,12,14)(-19,8,2,10)(5,15)(9,19)(13,17)
Multiloop annotated with half-edges
12^2_259 annotated with half-edges