$11^{1}_{66}$ - Minimal pinning sets
Pinning sets for 11^1_66 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_66 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, 6, 11}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 3, 4, 6}
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
3.23
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, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,6,3],[0,2,7,4],[1,3,7,8],[1,6,6,2],[2,5,5,8],[3,8,8,4],[4,7,7,6]]
PD code (use to draw this loop with SnapPy): [[7,18,8,1],[17,6,18,7],[8,11,9,12],[1,12,2,13],[13,16,14,17],[10,5,11,6],[9,5,10,4],[2,15,3,16],[14,3,15,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(2,7,-3,-8)(8,3,-9,-4)(15,4,-16,-5)(6,9,-7,-10)(17,10,-18,-11)(18,13,-1,-14)(11,14,-12,-15)(5,16,-6,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,14)(-2,-8,-4,15,-12)(-3,8)(-5,-17,-11,-15)(-6,-10,17)(-7,2,-13,18,10)(-9,6,16,4)(-14,11,-18)(-16,5)(1,13)(3,7,9)
Loop annotated with half-edges
11^1_66 annotated with half-edges