$12^{2}_{103}$ - Minimal pinning sets
Pinning sets for 12^2_103 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_103 Pinning data
Pinning number of this multiloop: 5
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: 2.97043
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 3, 4, 6, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
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
2.2
6
0
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 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],[0,8,8,1],[1,9,9,1],[2,9,8,7],[2,6,3,3],[4,6,9,4],[5,8,6,5]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,14,10,15],[4,7,5,8],[5,19,6,20],[1,13,2,14],[10,16,11,15],[17,3,18,4],[18,6,19,7],[12,2,13,3],[16,12,17,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(14,1,-15,-2)(15,4,-16,-5)(2,5,-3,-6)(20,13,-9,-14)(7,10,-8,-11)(18,11,-19,-12)(12,19,-13,-20)(3,16,-4,-17)(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)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-9)(-2,-6,-18,-12,-20,-14)(-3,-17,6)(-4,15,1,-10,7,17)(-5,2,-15)(-7,-11,18)(-8,9,13,19,11)(-13,20)(-16,3,5)(-19,12)(4,16)(8,10)
Multiloop annotated with half-edges
12^2_103 annotated with half-edges