$12^{1}_{135}$ - Minimal pinning sets
Pinning sets for 12^1_135 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_135 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)
• {2, 3, 6, 9, 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, 4, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,6],[0,7,7,8],[0,4,4,0],[1,3,3,8],[1,8,9,9],[1,9,9,7],[2,6,8,2],[2,7,5,4],[5,6,6,5]]
PD code (use to draw this loop with SnapPy): [[20,13,1,14],[14,3,15,4],[19,8,20,9],[12,1,13,2],[2,11,3,12],[15,7,16,6],[4,17,5,18],[9,18,10,19],[10,7,11,8],[16,5,17,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(14,3,-15,-4)(4,13,-5,-14)(8,5,-9,-6)(17,6,-18,-7)(20,9,-1,-10)(18,11,-19,-12)(2,15,-3,-16)(7,16,-8,-17)(10,19,-11,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-19,10)(-2,-16,7,-18,-12)(-3,14,-5,8,16)(-4,-14)(-6,17,-8)(-7,-17)(-9,20,-11,18,6)(-10,-20)(-13,4,-15,2)(1,9,5,13)(3,15)(11,19)
Loop annotated with half-edges
12^1_135 annotated with half-edges