$12^{2}_{123}$ - Minimal pinning sets
Pinning sets for 12^2_123
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_123
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, 5, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
•
{1, 2, 5, 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
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, 3, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,3],[0,2,6,7],[0,5,1,1],[1,4,8,8],[2,9,3,2],[3,9,9,8],[5,7,9,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[3,12,4,1],[2,20,3,13],[6,11,7,12],[4,7,5,8],[1,14,2,13],[14,19,15,20],[10,5,11,6],[8,17,9,16],[18,15,19,16],[9,17,10,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,3,-9,-4)(2,5,-3,-6)(4,9,-5,-10)(1,10,-2,-11)(19,16,-20,-17)(17,6,-18,-7)(7,18,-8,-19)(15,20,-16,-13)(12,13,-1,-14)(14,11,-15,-12)
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,-11,14)(-2,-6,17,-20,15,11)(-3,8,18,6)(-4,-10,1,13,-16,19,-8)(-5,2,10)(-7,-19,-17)(-9,4)(-12,-14)(-13,12,-15)(-18,7)(3,5,9)(16,20)
Multiloop annotated with half-edges
12^2_123 annotated with half-edges