$12^{1}_{31}$ - Minimal pinning sets
Pinning sets for 12^1_31 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_31 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, 2, 4, 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: [[1,2,2,3],[0,4,4,2],[0,1,5,0],[0,6,7,4],[1,3,5,1],[2,4,7,6],[3,5,8,8],[3,9,9,5],[6,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[8,11,9,12],[19,10,20,11],[1,6,2,7],[12,7,13,8],[13,18,14,19],[14,5,15,6],[2,17,3,18],[4,15,5,16],[16,3,17,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(17,2,-18,-3)(8,3,-9,-4)(15,6,-16,-7)(4,7,-5,-8)(20,11,-1,-12)(18,13,-19,-14)(9,14,-10,-15)(5,16,-6,-17)(12,19,-13,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,14,-19,12)(-2,17,-6,15,-10)(-3,8,-5,-17)(-4,-8)(-7,4,-9,-15)(-11,20,-13,18,2)(-12,-20)(-14,9,3,-18)(-16,5,7)(1,11)(6,16)(13,19)
Loop annotated with half-edges
12^1_31 annotated with half-edges