$12^{1}_{315}$ - Minimal pinning sets
Pinning sets for 12^1_315 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_315 Pinning data
Pinning number of this loop: 7
Total number of pinning sets: 32
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: 2.80821
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, 2, 3, 4, 6, 7, 11}
7
[2, 2, 2, 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
7
1
0
0
2.0
8
0
0
5
2.4
9
0
0
10
2.71
10
0
0
10
2.96
11
0
0
5
3.16
12
0
0
1
3.33
Total
1
0
31
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 2, 3, 4, 5, 7, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,4,5,5],[0,6,7,7],[1,7,2,1],[2,8,8,2],[3,9,9,7],[3,6,4,3],[5,9,9,5],[6,8,8,6]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[19,4,20,5],[6,18,7,17],[1,13,2,12],[3,18,4,19],[7,16,8,17],[13,10,14,11],[2,11,3,12],[15,8,16,9],[9,14,10,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (17,2,-18,-3)(13,6,-14,-7)(11,8,-12,-9)(20,9,-1,-10)(10,19,-11,-20)(7,12,-8,-13)(5,14,-6,-15)(15,4,-16,-5)(1,16,-2,-17)(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,-17,-3,-19,10)(-2,17)(-4,15,-6,13,-8,11,19)(-5,-15)(-7,-13)(-9,20,-11)(-10,-20)(-12,7,-14,5,-16,1,9)(-18,3)(2,16,4,18)(6,14)(8,12)
Loop annotated with half-edges
12^1_315 annotated with half-edges