$11^{1}_{34}$ - Minimal pinning sets
Pinning sets for 11^1_34 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_34 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 100
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: 2.9752
on average over minimal pinning sets: 2.49286
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)
• {1, 2, 4, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 6, 9, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 3, 5, 8, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 2, 3, 5, 6, 9, 10}
7
[2, 2, 2, 3, 3, 3, 3]
2.57
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
2
6
2.62
7
0
1
23
2.82
8
0
0
34
3.0
9
0
0
24
3.14
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
3
96
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,7,3],[0,2,7,7],[0,8,5,1],[1,4,8,1],[2,8,8,7],[2,6,3,3],[4,6,6,5]]
PD code (use to draw this loop with SnapPy): [[18,9,1,10],[10,4,11,3],[14,17,15,18],[15,8,16,9],[1,5,2,4],[11,2,12,3],[6,13,7,14],[7,16,8,17],[5,13,6,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,7,-1,-8)(1,16,-2,-17)(12,5,-13,-6)(15,6,-16,-7)(9,2,-10,-3)(3,10,-4,-11)(4,13,-5,-14)(11,14,-12,-15)(8,17,-9,-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,-17,8)(-2,9,17)(-3,-11,-15,-7,18,-9)(-4,-14,11)(-5,12,14)(-6,15,-12)(-8,-18)(-10,3)(-13,4,10,2,16,6)(-16,1,7)(5,13)
Loop annotated with half-edges
11^1_34 annotated with half-edges