$11^{1}_{68}$ - Minimal pinning sets
Pinning sets for 11^1_68 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_68 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 68
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91518
on average over minimal pinning sets: 2.39286
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 4, 7, 8, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {2, 3, 4, 6, 8, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 4, 6, 8, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
a (minimal)
• {1, 2, 4, 5, 7, 8, 10}
7
[2, 2, 2, 2, 3, 3, 4]
2.57
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.33
7
0
1
13
2.65
8
0
0
24
2.91
9
0
0
19
3.09
10
0
0
7
3.2
11
0
0
1
3.27
Total
3
1
64
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,6,6,0],[1,7,2,1],[2,7,8,8],[3,8,7,3],[4,6,8,5],[5,7,6,5]]
PD code (use to draw this loop with SnapPy): [[9,18,10,1],[8,15,9,16],[17,14,18,15],[10,2,11,1],[16,7,17,8],[4,13,5,14],[2,12,3,11],[3,6,4,7],[12,5,13,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,3,-11,-4)(1,4,-2,-5)(14,5,-15,-6)(9,18,-10,-1)(2,11,-3,-12)(15,12,-16,-13)(6,13,-7,-14)(7,16,-8,-17)(17,8,-18,-9)
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,-5,14,-7,-17,-9)(-2,-12,15,5)(-3,10,18,8,16,12)(-4,1,-10)(-6,-14)(-8,17)(-11,2,4)(-13,6,-15)(-16,7,13)(-18,9)(3,11)
Loop annotated with half-edges
11^1_68 annotated with half-edges