$12^{2}_{233}$ - Minimal pinning sets
Pinning sets for 12^2_233 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_233 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, 2, 4, 9, 10}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 8, 9}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 4, 7, 9}
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, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,3,3],[0,2,2,7],[0,8,8,5],[1,4,6,1],[1,5,9,2],[3,9,9,8],[4,7,9,4],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[9,12,10,1],[8,20,9,13],[15,11,16,12],[10,16,11,17],[1,6,2,7],[13,7,14,8],[14,19,15,20],[17,4,18,3],[5,2,6,3],[18,4,19,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,12,-4,-1)(1,20,-2,-13)(13,2,-14,-3)(11,4,-12,-5)(16,5,-17,-6)(18,9,-19,-10)(7,10,-8,-11)(8,19,-9,-20)(17,14,-18,-15)(6,15,-7,-16)
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,-3)(-2,13)(-4,11,-8,-20,1)(-5,16,-7,-11)(-6,-16)(-9,18,14,2,20)(-10,7,15,-18)(-12,3,-14,17,5)(-15,6,-17)(-19,8,10)(4,12)(9,19)
Multiloop annotated with half-edges
12^2_233 annotated with half-edges