$12^{1}_{131}$ - Minimal pinning sets
Pinning sets for 12^1_131 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_131 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 240
of which optimal: 2
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03455
on average over minimal pinning sets: 2.45
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 7, 10}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 6, 7, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 5, 7, 9, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 3, 6, 7, 9, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.4
6
0
2
13
2.66
7
0
0
45
2.86
8
0
0
71
3.02
9
0
0
64
3.14
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
2
236
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,3,0],[0,2,6,1],[1,7,7,5],[1,4,7,8],[2,8,9,3],[4,9,5,4],[5,9,9,6],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,18,9,17],[6,19,7,20],[1,19,2,18],[9,15,10,14],[16,13,17,14],[5,2,6,3],[15,11,16,10],[12,3,13,4],[4,11,5,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,20,-4,-1)(10,1,-11,-2)(18,5,-19,-6)(8,13,-9,-14)(2,9,-3,-10)(14,11,-15,-12)(12,7,-13,-8)(15,6,-16,-7)(16,19,-17,-20)(4,17,-5,-18)
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,-3)(-2,-10)(-4,-18,-6,15,11,1)(-5,18)(-7,12,-15)(-8,-14,-12)(-9,2,-11,14)(-13,8)(-16,-20,3,9,13,7)(-17,4,20)(-19,16,6)(5,17,19)
Loop annotated with half-edges
12^1_131 annotated with half-edges