$12^{2}_{240}$ - Minimal pinning sets
Pinning sets for 12^2_240 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_240 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, 2, 5, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 5, 8, 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
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, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,6],[0,6,7,7],[0,7,7,8],[0,9,9,5],[1,4,9,8],[1,8,2,1],[2,3,3,2],[3,6,5,9],[4,8,5,4]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,6,16,5],[17,13,18,14],[19,11,20,12],[1,8,2,9],[6,9,7,10],[16,4,17,5],[12,18,13,19],[10,3,11,4],[7,2,8,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,14,-8,-1)(16,1,-17,-2)(2,15,-3,-16)(12,5,-13,-6)(3,6,-4,-7)(19,8,-20,-9)(17,10,-18,-11)(4,13,-5,-14)(9,18,-10,-19)(11,20,-12,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-3,-7)(-2,-16)(-4,-14,7)(-5,12,20,8,14)(-6,3,15,-12)(-8,19,-10,17,1)(-9,-19)(-11,-15,2,-17)(-13,4,6)(-18,9,-20,11)(5,13)(10,18)
Multiloop annotated with half-edges
12^2_240 annotated with half-edges