$12^{1}_{286}$ - Minimal pinning sets
Pinning sets for 12^1_286 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_286 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 512
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10608
on average over minimal pinning sets: 2.5
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 11}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 3, 8, 11}
4
[2, 2, 3, 3]
2.50
C (optimal)
• {1, 3, 5, 11}
4
[2, 2, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
3
0
0
2.5
5
0
0
22
2.76
6
0
0
70
2.94
7
0
0
126
3.06
8
0
0
140
3.15
9
0
0
98
3.22
10
0
0
42
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
3
0
509
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,7],[0,7,7,8],[0,8,8,5],[1,4,9,6],[1,5,9,2],[2,9,3,3],[3,9,4,4],[5,8,7,6]]
PD code (use to draw this loop with SnapPy): [[20,11,1,12],[12,7,13,8],[8,19,9,20],[10,3,11,4],[1,16,2,17],[17,6,18,7],[13,18,14,19],[9,5,10,4],[15,2,16,3],[5,14,6,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,20,-14,-1)(8,1,-9,-2)(2,7,-3,-8)(15,4,-16,-5)(10,5,-11,-6)(18,9,-19,-10)(3,12,-4,-13)(19,14,-20,-15)(11,16,-12,-17)(6,17,-7,-18)
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,-3,-13)(-2,-8)(-4,15,-20,13)(-5,10,-19,-15)(-6,-18,-10)(-7,2,-9,18)(-11,-17,6)(-12,3,7,17)(-14,19,9,1)(-16,11,5)(4,12,16)(14,20)
Loop annotated with half-edges
12^1_286 annotated with half-edges