$12^{2}_{11}$ - Minimal pinning sets
Pinning sets for 12^2_11 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_11 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 240
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: 2.97778
on average over minimal pinning sets: 2.3
on average over optimal pinning sets: 2.3
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 7, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 5, 8, 9}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 3, 5, 9, 12}
5
[2, 2, 2, 2, 3]
2.20
D (optimal)
• {1, 3, 5, 9, 11}
5
[2, 2, 2, 2, 4]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.3
6
0
0
22
2.62
7
0
0
52
2.84
8
0
0
69
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
236
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: [[1,2,3,4],[0,4,4,5],[0,6,6,7],[0,7,8,8],[0,5,1,1],[1,4,9,9],[2,9,7,2],[2,6,8,3],[3,7,9,3],[5,8,6,5]]
PD code (use to draw this multiloop with SnapPy): [[3,14,4,1],[9,2,10,3],[13,20,14,15],[4,17,5,18],[1,8,2,9],[10,8,11,7],[15,12,16,13],[16,19,17,20],[5,19,6,18],[11,6,12,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,14,-8,-1)(9,4,-10,-5)(5,8,-6,-9)(13,6,-14,-7)(19,10,-20,-11)(1,12,-2,-13)(2,15,-3,-16)(16,3,-17,-4)(20,17,-15,-18)(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,-13,-7)(-2,-16,-4,9,-6,13)(-3,16)(-5,-9)(-8,5,-10,19,-12,1)(-11,-19)(-14,7)(-15,2,12,18)(-17,20,10,4)(-18,11,-20)(3,15,17)(6,8,14)
Multiloop annotated with half-edges
12^2_11 annotated with half-edges