$11^{2}_{72}$ - Minimal pinning sets
Pinning sets for 11^2_72 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_72 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 160
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97092
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 7, 10}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 6, 9, 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
4
1
0
0
2.25
5
0
1
7
2.55
6
0
0
26
2.77
7
0
0
45
2.93
8
0
0
45
3.06
9
0
0
26
3.15
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
1
158
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,6,6,7],[0,7,7,1],[1,8,8,2],[2,8,3,3],[3,8,4,4],[5,7,6,5]]
PD code (use to draw this multiloop with SnapPy): [[8,18,1,9],[9,14,10,15],[15,7,16,8],[17,3,18,4],[1,13,2,14],[10,6,11,7],[16,5,17,4],[12,2,13,3],[5,11,6,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(14,1,-15,-2)(2,13,-3,-14)(17,4,-18,-5)(12,5,-13,-6)(3,18,-4,-9)(7,10,-8,-11)(16,11,-17,-12)(6,15,-7,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-3,-9)(-2,-14)(-4,17,11,-8,9)(-5,12,-17)(-6,-16,-12)(-7,-11,16)(-10,7,15,1)(-13,2,-15,6)(-18,3,13,5)(4,18)(8,10)
Multiloop annotated with half-edges
11^2_72 annotated with half-edges