$11^{1}_{39}$ - Minimal pinning sets
Pinning sets for 11^1_39 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_39 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 144
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97268
on average over minimal pinning sets: 2.4
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, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 4, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 3, 4, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.4
6
0
0
20
2.7
7
0
0
41
2.91
8
0
0
44
3.05
9
0
0
26
3.15
10
0
0
8
3.23
11
0
0
1
3.27
Total
4
0
140
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: [[1,2,3,4],[0,4,5,6],[0,7,8,8],[0,8,4,4],[0,3,3,1],[1,8,7,6],[1,5,7,7],[2,6,6,5],[2,5,3,2]]
PD code (use to draw this loop with SnapPy): [[18,3,1,4],[4,14,5,13],[17,10,18,11],[2,15,3,16],[1,15,2,14],[5,9,6,8],[12,7,13,8],[11,7,12,6],[9,16,10,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(4,9,-5,-10)(10,5,-11,-6)(6,3,-7,-4)(15,8,-16,-9)(11,2,-12,-3)(12,17,-13,-18)(18,13,-1,-14)(7,14,-8,-15)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,8,14)(-2,11,5,9,-16)(-3,6,-11)(-4,-10,-6)(-5,10)(-7,-15,-9,4)(-8,15)(-12,-18,-14,7,3)(-13,18)(-17,12,2)(1,13,17)
Loop annotated with half-edges
11^1_39 annotated with half-edges