$12^{2}_{200}$ - Minimal pinning sets
Pinning sets for 12^2_200 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_200 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, 3, 5, 6, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 5, 6, 9}
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, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,5],[0,6,7,4],[1,3,7,8],[2,6,6,2],[3,5,5,9],[3,9,8,4],[4,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,7,10,8],[10,19,11,20],[1,13,2,14],[14,6,15,7],[18,11,19,12],[12,17,13,18],[2,5,3,6],[15,3,16,4],[4,16,5,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-9,-2)(8,3,-1,-4)(19,4,-20,-5)(14,5,-15,-6)(11,16,-12,-17)(7,18,-8,-19)(2,9,-3,-10)(17,10,-18,-11)(15,12,-16,-13)(6,13,-7,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,4)(-2,-10,17,-12,15,5,-20)(-3,8,18,10)(-4,19,-8)(-5,14,-7,-19)(-6,-14)(-9,2)(-11,-17)(-13,6,-15)(-16,11,-18,7,13)(1,3,9)(12,16)
Multiloop annotated with half-edges
12^2_200 annotated with half-edges