$11^{1}_{65}$ - Minimal pinning sets
Pinning sets for 11^1_65 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_65 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96934
on average over minimal pinning sets: 2.25
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, 8, 9}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 3, 8, 10}
4
[2, 2, 2, 3]
2.25
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.25
5
0
0
13
2.58
6
0
0
36
2.81
7
0
0
55
2.96
8
0
0
50
3.08
9
0
0
27
3.16
10
0
0
8
3.22
11
0
0
1
3.27
Total
2
0
190
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],[0,5,6,6],[0,6,3,3],[0,2,2,7],[0,8,8,5],[1,4,8,7],[1,7,2,1],[3,6,5,8],[4,7,5,4]]
PD code (use to draw this loop with SnapPy): [[15,18,16,1],[14,5,15,6],[17,12,18,13],[16,12,17,11],[1,8,2,9],[6,9,7,10],[4,13,5,14],[10,3,11,4],[7,2,8,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,18,-8,-1)(11,2,-12,-3)(16,5,-17,-6)(3,6,-4,-7)(8,13,-9,-14)(14,9,-15,-10)(1,10,-2,-11)(12,15,-13,-16)(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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-11,-3,-7)(-2,11)(-4,-18,7)(-5,16,-13,8,18)(-6,3,-12,-16)(-8,-14,-10,1)(-9,14)(-15,12,2,10)(-17,4,6)(5,17)(9,13,15)
Loop annotated with half-edges
11^1_65 annotated with half-edges