$12^{2}_{197}$ - Minimal pinning sets
Pinning sets for 12^2_197 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_197 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 176
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97631
on average over minimal pinning sets: 2.34444
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)
• {2, 3, 6, 9, 10}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {2, 3, 6, 8, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
b (minimal)
• {1, 2, 3, 6, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
2
7
2.5
7
0
0
30
2.75
8
0
0
51
2.94
9
0
0
49
3.08
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
2
173
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,5],[0,6,4,7],[0,7,7,0],[1,7,2,8],[1,8,9,1],[2,9,9,8],[2,4,3,3],[4,6,9,5],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,10,14,9],[17,11,18,12],[19,1,20,2],[10,3,11,4],[14,8,15,9],[6,16,7,17],[18,3,19,2],[4,7,5,8],[15,5,16,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,13,-1,-14)(8,1,-9,-2)(18,3,-19,-4)(2,5,-3,-6)(17,6,-18,-7)(7,16,-8,-17)(20,9,-13,-10)(14,11,-15,-12)(4,19,-5,-20)(10,15,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,16,-11,14)(-2,-6,17,-8)(-3,18,6)(-4,-20,-10,-16,7,-18)(-5,2,-9,20)(-7,-17)(-12,-14)(-13,12,-15,10)(-19,4)(1,13,9)(3,5,19)(11,15)
Multiloop annotated with half-edges
12^2_197 annotated with half-edges