$12^{2}_{69}$ - Minimal pinning sets
Pinning sets for 12^2_69 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_69 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, 5, 7, 11}
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, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,6,6,1],[2,7,8,8],[2,8,9,6],[3,5,7,3],[4,6,9,9],[4,9,5,4],[5,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[16,7,1,8],[8,15,9,16],[9,6,10,7],[1,14,2,15],[5,20,6,17],[10,4,11,3],[13,2,14,3],[17,13,18,12],[19,4,20,5],[11,19,12,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,16,-12,-1)(1,8,-2,-9)(17,2,-18,-3)(7,4,-8,-5)(14,5,-15,-6)(10,19,-11,-20)(15,12,-16,-13)(6,13,-7,-14)(20,9,-17,-10)(3,18,-4,-19)
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,-9,20,-11)(-2,17,9)(-3,-19,10,-17)(-4,7,13,-16,11,19)(-5,14,-7)(-6,-14)(-8,1,-12,15,5)(-10,-20)(-13,6,-15)(-18,3)(2,8,4,18)(12,16)
Multiloop annotated with half-edges
12^2_69 annotated with half-edges