$11^{1}_{70}$ - Minimal pinning sets
Pinning sets for 11^1_70 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_70 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 104
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97496
on average over minimal pinning sets: 2.475
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 5, 6, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 5, 6, 10, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 4, 5, 6, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 4, 5, 6, 10, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
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
2.4
6
0
3
6
2.61
7
0
0
26
2.84
8
0
0
35
3.01
9
0
0
24
3.14
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
3
100
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,5,6],[0,6,6,0],[0,7,8,8],[1,8,7,5],[1,4,7,6],[1,5,2,2],[3,5,4,8],[3,7,4,3]]
PD code (use to draw this loop with SnapPy): [[18,7,1,8],[8,15,9,16],[6,17,7,18],[1,13,2,12],[3,14,4,15],[9,4,10,5],[16,5,17,6],[13,10,14,11],[2,11,3,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,2,-8,-3)(14,3,-15,-4)(16,5,-17,-6)(11,8,-12,-9)(18,9,-1,-10)(10,17,-11,-18)(1,12,-2,-13)(6,13,-7,-14)(4,15,-5,-16)
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,-13,6,-17,10)(-2,7,13)(-3,14,-7)(-4,-16,-6,-14)(-5,16)(-8,11,17,5,15,3)(-9,18,-11)(-10,-18)(-12,1,9)(-15,4)(2,12,8)
Loop annotated with half-edges
11^1_70 annotated with half-edges