$12^{1}_{278}$ - Minimal pinning sets
Pinning sets for 12^1_278 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_278 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 80
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91429
on average over minimal pinning sets: 2.22619
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 7, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
a (minimal)
• {1, 2, 3, 4, 5, 6, 9}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
1
0
0
2.17
7
0
1
6
2.47
8
0
0
19
2.73
9
0
0
26
2.94
10
0
0
19
3.12
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
1
78
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 3, 5, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,3],[0,2,6,6],[0,7,8,8],[1,8,2,1],[3,9,9,3],[4,9,9,8],[4,7,5,4],[6,7,7,6]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[4,17,5,18],[19,16,20,17],[6,16,7,15],[1,10,2,11],[18,3,19,4],[7,14,8,15],[9,12,10,13],[2,12,3,11],[13,8,14,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(10,5,-11,-6)(18,7,-19,-8)(8,19,-9,-20)(6,9,-7,-10)(4,11,-5,-12)(12,3,-13,-4)(16,13,-17,-14)(1,14,-2,-15)(2,17,-3,-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,-15)(-2,-18,-8,-20,15)(-3,12,-5,10,-7,18)(-4,-12)(-6,-10)(-9,6,-11,4,-13,16,20)(-14,1,-16)(-17,2,14)(-19,8)(3,17,13)(5,11)(7,9,19)
Loop annotated with half-edges
12^1_278 annotated with half-edges