$11^{1}_{31}$ - Minimal pinning sets
Pinning sets for 11^1_31 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_31 Pinning data
Pinning number of this loop: 5
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: 2.90697
on average over minimal pinning sets: 2.26667
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)
• {1, 3, 4, 6, 10}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 5, 6, 10}
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
1
6
2.5
7
0
0
19
2.74
8
0
0
26
2.94
9
0
0
19
3.09
10
0
0
7
3.2
11
0
0
1
3.27
Total
1
1
78
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 3, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,3,4],[0,5,6,0],[0,4,1,1],[1,3,7,7],[2,7,8,6],[2,5,8,8],[4,8,5,4],[5,7,6,6]]
PD code (use to draw this loop with SnapPy): [[3,18,4,1],[2,11,3,12],[17,4,18,5],[1,13,2,12],[13,10,14,11],[5,9,6,8],[16,7,17,8],[9,14,10,15],[6,15,7,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(14,5,-15,-6)(6,3,-7,-4)(11,8,-12,-9)(18,9,-1,-10)(10,17,-11,-18)(7,12,-8,-13)(4,13,-5,-14)(1,16,-2,-17)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-17,10)(-2,15,5,13,-8,11,17)(-3,6,-15)(-4,-14,-6)(-5,14)(-7,-13,4)(-9,18,-11)(-10,-18)(-12,7,3,-16,1,9)(2,16)(8,12)
Loop annotated with half-edges
11^1_31 annotated with half-edges