$11^{1}_{63}$ - Minimal pinning sets
Pinning sets for 11^1_63 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_63 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 112
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91313
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 10}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 4, 9, 10}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 3, 4, 8, 10}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.27
6
0
0
15
2.6
7
0
0
31
2.83
8
0
0
34
2.99
9
0
0
21
3.11
10
0
0
7
3.2
11
0
0
1
3.27
Total
3
0
109
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,6,6,0],[0,7,8,4],[1,3,5,1],[1,4,8,6],[2,5,7,2],[3,6,8,8],[3,7,7,5]]
PD code (use to draw this loop with SnapPy): [[7,18,8,1],[6,13,7,14],[17,8,18,9],[1,4,2,5],[14,5,15,6],[15,12,16,13],[9,16,10,17],[10,3,11,4],[2,11,3,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,18,-8,-1)(16,1,-17,-2)(11,2,-12,-3)(14,5,-15,-6)(17,8,-18,-9)(12,9,-13,-10)(3,10,-4,-11)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-5,14,-7)(-2,11,-4,-16)(-3,-11)(-6,-14)(-8,17,1)(-9,12,2,-17)(-10,3,-12)(-13,6,-15,4,10)(-18,7,13,9)(5,15)(8,18)
Loop annotated with half-edges
11^1_63 annotated with half-edges