$11^{1}_{76}$ - Minimal pinning sets
Pinning sets for 11^1_76 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_76 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)
• {1, 2, 3, 5, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 4, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 3, 5, 7, 8}
6
[2, 2, 2, 2, 3, 3]
2.33
a (minimal)
• {1, 2, 3, 4, 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, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,5,3],[0,2,6,0],[1,7,7,1],[2,7,8,6],[3,5,8,8],[4,8,5,4],[5,7,6,6]]
PD code (use to draw this loop with SnapPy): [[18,9,1,10],[10,4,11,3],[17,2,18,3],[8,1,9,2],[4,12,5,11],[13,16,14,17],[14,7,15,8],[12,6,13,5],[6,15,7,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,18,-16,-1)(1,8,-2,-9)(3,10,-4,-11)(12,5,-13,-6)(16,7,-17,-8)(9,2,-10,-3)(4,13,-5,-14)(11,14,-12,-15)(6,17,-7,-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,-9,-3,-11,-15)(-2,9)(-4,-14,11)(-5,12,14)(-6,-18,15,-12)(-7,16,18)(-8,1,-16)(-10,3)(-13,4,10,2,8,-17,6)(5,13)(7,17)
Loop annotated with half-edges
11^1_76 annotated with half-edges