$12^{2}_{127}$ - Minimal pinning sets
Pinning sets for 12^2_127 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_127 Pinning data
Pinning number of this multiloop: 7
Total number of pinning sets: 60
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.86652
on average over minimal pinning sets: 2.21429
on average over optimal pinning sets: 2.21429
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 5, 7, 10}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
B (optimal)
• {1, 2, 3, 4, 5, 7, 12}
7
[2, 2, 2, 2, 2, 2, 4]
2.29
C (optimal)
• {1, 2, 3, 4, 5, 7, 11}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
D (optimal)
• {1, 2, 3, 4, 5, 7, 9}
7
[2, 2, 2, 2, 2, 2, 4]
2.29
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
7
4
0
0
2.21
8
0
0
14
2.62
9
0
0
20
2.89
10
0
0
15
3.07
11
0
0
6
3.21
12
0
0
1
3.33
Total
4
0
56
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 7, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,5,6,6],[0,7,1,0],[1,7,5,5],[1,4,4,2],[2,8,8,2],[3,9,9,4],[6,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,19,16,18],[13,6,14,7],[19,1,20,2],[16,9,17,8],[17,7,18,8],[5,12,6,13],[2,10,3,9],[11,4,12,5],[10,4,11,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,7,-1,-8)(8,1,-9,-2)(10,3,-11,-4)(15,4,-16,-5)(5,20,-6,-15)(6,13,-7,-14)(2,9,-3,-10)(17,12,-18,-13)(16,19,-17,-20)(11,18,-12,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,-10,-4,15,-6,-14,-8)(-3,10)(-5,-15)(-7,14)(-9,2)(-11,-19,16,4)(-12,17,19)(-13,6,20,-17)(-16,-20,5)(-18,11,3,9,1,7,13)(12,18)
Multiloop annotated with half-edges
12^2_127 annotated with half-edges