$12^{1}_{6}$ - Minimal pinning sets
Pinning sets for 12^1_6 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_6 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 128
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: 2.90623
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, 3, 6, 7, 11}
5
[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
5
1
0
0
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,5,5,6],[0,6,7,7],[1,7,8,1],[2,8,9,2],[2,9,7,3],[3,6,4,3],[4,9,9,5],[5,8,8,6]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[19,4,20,5],[6,11,7,12],[1,15,2,14],[3,18,4,19],[10,7,11,8],[12,16,13,15],[2,13,3,14],[17,8,18,9],[9,16,10,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (17,2,-18,-3)(8,5,-9,-6)(15,6,-16,-7)(4,9,-5,-10)(20,11,-1,-12)(12,19,-13,-20)(13,10,-14,-11)(7,14,-8,-15)(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,-19,12)(-2,17)(-4,-10,13,19)(-5,8,14,10)(-6,15,-8)(-7,-15)(-9,4,18,2,16,6)(-11,20,-13)(-12,-20)(-14,7,-16,1,11)(-18,3)(5,9)
Loop annotated with half-edges
12^1_6 annotated with half-edges