$12^{2}_{76}$ - Minimal pinning sets
Pinning sets for 12^2_76 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_76 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, 4, 5, 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],[0,3,2,0],[0,1,4,5],[0,6,7,1],[2,7,8,8],[2,9,6,6],[3,5,5,7],[3,6,9,4],[4,9,9,4],[5,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[14,7,1,8],[8,13,9,14],[9,6,10,7],[1,12,2,13],[5,20,6,15],[10,18,11,17],[11,16,12,17],[2,16,3,15],[19,4,20,5],[18,4,19,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,14,-16,-1)(5,2,-6,-3)(10,3,-11,-4)(11,8,-12,-9)(4,9,-5,-10)(13,20,-14,-15)(7,18,-8,-19)(19,6,-20,-7)(1,16,-2,-17)(17,12,-18,-13)
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,-17,-13,-15)(-2,5,9,-12,17)(-3,10,-5)(-4,-10)(-6,19,-8,11,3)(-7,-19)(-9,4,-11)(-14,15)(-16,1)(-18,7,-20,13)(2,16,14,20,6)(8,18,12)
Multiloop annotated with half-edges
12^2_76 annotated with half-edges