$12^{2}_{46}$ - Minimal pinning sets
Pinning sets for 12^2_46 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_46 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 112
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: 2.91983
on average over minimal pinning sets: 2.22222
on average over optimal pinning sets: 2.22222
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 6, 7, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 7, 10, 11}
6
[2, 2, 2, 2, 2, 4]
2.33
C (optimal)
• {1, 2, 3, 7, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.22
7
0
0
15
2.57
8
0
0
31
2.82
9
0
0
34
3.01
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
3
0
109
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,7,7],[0,8,8,6],[0,9,9,1],[1,9,6,1],[2,5,9,3],[2,8,8,2],[3,7,7,3],[4,6,5,4]]
PD code (use to draw this multiloop with SnapPy): [[11,16,12,1],[10,20,11,17],[4,15,5,16],[12,7,13,8],[1,18,2,17],[19,9,20,10],[3,8,4,9],[14,5,15,6],[6,13,7,14],[18,3,19,2]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(9,4,-10,-5)(16,5,-1,-6)(17,6,-18,-7)(3,10,-4,-11)(1,12,-2,-13)(8,13,-9,-14)(15,20,-16,-17)(7,18,-8,-19)(19,14,-20,-15)
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,-13,8,18,6)(-2,11,-4,9,13)(-3,-11)(-5,16,20,14,-9)(-6,17,-16)(-7,-19,-15,-17)(-8,-14,19)(-10,3,-12,1,5)(-18,7)(-20,15)(2,12)(4,10)
Multiloop annotated with half-edges
12^2_46 annotated with half-edges