$12^{2}_{105}$ - Minimal pinning sets
Pinning sets for 12^2_105 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_105 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 224
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: 2.9785
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, 5, 7, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 5, 7, 10, 11}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 5, 6, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.27
6
0
0
18
2.59
7
0
0
46
2.82
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
221
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,6,3],[0,2,7,7],[0,8,9,1],[1,9,9,1],[2,8,7,2],[3,6,8,3],[4,7,6,9],[4,8,5,5]]
PD code (use to draw this multiloop with SnapPy): [[5,16,6,1],[4,11,5,12],[15,20,16,17],[6,20,7,19],[1,13,2,12],[10,3,11,4],[17,14,18,15],[7,18,8,19],[13,8,14,9],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,16,-8,-1)(10,5,-11,-6)(15,6,-16,-7)(1,8,-2,-9)(9,14,-10,-15)(20,11,-17,-12)(2,13,-3,-14)(4,17,-5,-18)(18,3,-19,-4)(12,19,-13,-20)
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,-9,-15,-7)(-2,-14,9)(-3,18,-5,10,14)(-4,-18)(-6,15,-10)(-8,1)(-11,20,-13,2,8,16,6)(-12,-20)(-16,7)(-17,4,-19,12)(3,13,19)(5,17,11)
Multiloop annotated with half-edges
12^2_105 annotated with half-edges