$12^{2}_{243}$ - Minimal pinning sets
Pinning sets for 12^2_243 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_243 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96564
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 7, 9}
4
[2, 2, 2, 2]
2.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
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: [[1,1,2,3],[0,4,5,0],[0,5,6,3],[0,2,7,8],[1,8,8,9],[1,6,6,2],[2,5,5,9],[3,9,9,8],[3,7,4,4],[4,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[7,16,8,1],[15,6,16,7],[8,11,9,12],[1,12,2,13],[14,20,15,17],[10,5,11,6],[9,5,10,4],[2,19,3,18],[13,18,14,17],[3,19,4,20]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,6,-2,-7)(7,2,-8,-3)(12,3,-13,-4)(5,8,-6,-9)(14,9,-15,-10)(4,13,-5,-14)(20,15,-17,-16)(16,17,-1,-18)(11,18,-12,-19)(19,10,-20,-11)
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,-7,-3,12,18)(-2,7)(-4,-14,-10,19,-12)(-5,-9,14)(-6,1,17,15,9)(-8,5,13,3)(-11,-19)(-13,4)(-15,20,10)(-16,-18,11,-20)(-17,16)(2,6,8)
Multiloop annotated with half-edges
12^2_243 annotated with half-edges