$12^{1}_{15}$ - Minimal pinning sets
Pinning sets for 12^1_15 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_15 Pinning data
Pinning number of this loop: 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, 2, 3, 11}
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 loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,7],[0,7,8,8],[0,5,1,1],[1,4,8,2],[2,8,9,9],[2,9,9,3],[3,6,5,3],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,7,3,8],[10,19,11,20],[4,16,5,15],[1,9,2,8],[9,6,10,7],[18,13,19,14],[11,17,12,16],[5,14,6,15],[12,17,13,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,4,-8,-5)(20,5,-1,-6)(6,19,-7,-20)(16,11,-17,-12)(3,12,-4,-13)(13,2,-14,-3)(14,9,-15,-10)(10,15,-11,-16)(8,17,-9,-18)(1,18,-2,-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,-19,6)(-2,13,-4,7,19)(-3,-13)(-5,20,-7)(-6,-20)(-8,-18,1,5)(-9,14,2,18)(-10,-16,-12,3,-14)(-11,16)(-15,10)(-17,8,4,12)(9,17,11,15)
Loop annotated with half-edges
12^1_15 annotated with half-edges