$12^{1}_{364}$ - Minimal pinning sets
Pinning sets for 12^1_364 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_364 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 184
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97614
on average over minimal pinning sets: 2.38333
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 6, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {2, 3, 5, 7, 10, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
b (minimal)
• {2, 3, 5, 7, 9, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
c (minimal)
• {1, 2, 3, 5, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
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
2.2
6
0
3
7
2.5
7
0
0
33
2.76
8
0
0
54
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
3
180
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,6],[0,7,7,3],[0,2,8,4],[0,3,5,5],[1,4,4,8],[1,8,9,1],[2,9,9,2],[3,9,6,5],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,3,9,4],[12,19,13,20],[13,6,14,7],[1,14,2,15],[15,2,16,3],[9,5,10,4],[18,11,19,12],[5,16,6,17],[10,17,11,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,5,-1,-6)(9,2,-10,-3)(16,3,-17,-4)(14,7,-15,-8)(17,10,-18,-11)(1,12,-2,-13)(6,13,-7,-14)(8,15,-9,-16)(11,18,-12,-19)(4,19,-5,-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,-13,6)(-2,9,15,7,13)(-3,16,-9)(-4,-20,-6,-14,-8,-16)(-5,20)(-7,14)(-10,17,3)(-11,-19,4,-17)(-12,1,5,19)(-15,8)(-18,11)(2,12,18,10)
Loop annotated with half-edges
12^1_364 annotated with half-edges