$12^{2}_{311}$ - Minimal pinning sets
Pinning sets for 12^2_311 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_311 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 320
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03463
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 6, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 5, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
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
2.25
5
0
1
8
2.56
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
318
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,6,7,3],[0,2,7,4],[1,3,7,8],[1,8,6,6],[2,5,5,9],[2,9,4,3],[4,9,9,5],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[11,16,12,1],[15,10,16,11],[12,8,13,7],[1,7,2,6],[14,5,15,6],[9,20,10,17],[8,20,9,19],[13,3,14,2],[4,17,5,18],[18,3,19,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (17,16,-18,-1)(11,2,-12,-3)(3,14,-4,-15)(4,9,-5,-10)(12,7,-13,-8)(8,13,-9,-14)(15,10,-16,-11)(18,5,-19,-6)(6,19,-7,-20)(1,20,-2,-17)
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)(-2,11,-16,17)(-3,-15,-11)(-4,-10,15)(-5,18,16,10)(-6,-20,1,-18)(-7,12,2,20)(-8,-14,3,-12)(-9,4,14)(-13,8)(-19,6)(5,9,13,7,19)
Multiloop annotated with half-edges
12^2_311 annotated with half-edges