$11^{1}_{52}$ - Minimal pinning sets
Pinning sets for 11^1_52
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_52
Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 176
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97775
on average over minimal pinning sets: 2.41667
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 3, 7, 10}
4
[2, 2, 2, 3]
2.25
a (minimal)
•
{1, 2, 3, 5, 10}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
•
{1, 3, 5, 6, 10}
5
[2, 2, 2, 3, 4]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
2
7
2.56
6
0
0
30
2.78
7
0
0
51
2.95
8
0
0
49
3.07
9
0
0
27
3.16
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
2
173
Other information about this loop
Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,7],[0,7,8,8],[1,8,2,1],[2,8,7,3],[3,6,4,3],[4,6,5,4]]
PD code (use to draw this loop with SnapPy): [[5,18,6,1],[4,15,5,16],[17,14,18,15],[6,11,7,12],[1,8,2,9],[16,3,17,4],[10,13,11,14],[7,13,8,12],[2,10,3,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(17,6,-18,-7)(7,18,-8,-1)(4,9,-5,-10)(10,5,-11,-6)(8,11,-9,-12)(15,12,-16,-13)(2,13,-3,-14)(3,16,-4,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-3,-17,-7)(-2,-14)(-4,-10,-6,17)(-5,10)(-8,-12,15,1)(-9,4,16,12)(-11,8,18,6)(-13,2,-15)(-16,3,13)(-18,7)(5,9,11)
Loop annotated with half-edges
11^1_52 annotated with half-edges