$12^{2}_{223}$ - Minimal pinning sets
Pinning sets for 12^2_223 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_223 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 6, 7}
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
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,7,7,4],[0,3,8,5],[1,4,2,1],[2,8,8,9],[3,9,9,3],[4,9,6,6],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,6,10,5],[7,4,8,5],[19,12,20,13],[1,12,2,11],[6,11,7,10],[3,15,4,16],[13,18,14,19],[2,17,3,16],[17,14,18,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,8,-4,-1)(1,20,-2,-9)(9,2,-10,-3)(10,5,-11,-6)(18,11,-19,-12)(16,13,-17,-14)(7,14,-8,-15)(15,6,-16,-7)(12,17,-13,-18)(4,19,-5,-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,-3)(-2,9)(-4,-20,1)(-5,10,2,20)(-6,15,-8,3,-10)(-7,-15)(-11,18,-13,16,6)(-12,-18)(-14,7,-16)(-17,12,-19,4,8,14)(5,19,11)(13,17)
Multiloop annotated with half-edges
12^2_223 annotated with half-edges