$12^{2}_{10}$ - Minimal pinning sets
Pinning sets for 12^2_10 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_10 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96564
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 6, 9}
4
[2, 2, 2, 2]
2.00
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
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,7],[0,8,9,9],[0,5,1,1],[1,4,9,6],[2,5,7,2],[2,6,8,8],[3,7,7,9],[3,8,5,3]]
PD code (use to draw this multiloop with SnapPy): [[3,12,4,1],[2,7,3,8],[11,20,12,13],[4,18,5,17],[1,9,2,8],[9,6,10,7],[13,10,14,11],[14,19,15,20],[18,15,19,16],[5,16,6,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,4,-8,-5)(12,5,-1,-6)(6,11,-7,-12)(17,8,-18,-9)(1,10,-2,-11)(14,19,-15,-20)(3,20,-4,-13)(13,2,-14,-3)(18,15,-19,-16)(9,16,-10,-17)
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,-11,6)(-2,13,-4,7,11)(-3,-13)(-5,12,-7)(-6,-12)(-8,17,-10,1,5)(-9,-17)(-14,-20,3)(-15,18,8,4,20)(-16,9,-18)(-19,14,2,10,16)(15,19)
Multiloop annotated with half-edges
12^2_10 annotated with half-edges