$12^{2}_{302}$ - Minimal pinning sets
Pinning sets for 12^2_302 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_302 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 288
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: 3.03457
on average over minimal pinning sets: 2.375
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, 4, 6, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 5, 6, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
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
0
8
2.57
6
0
1
28
2.78
7
0
0
61
2.93
8
0
0
80
3.05
9
0
0
66
3.15
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
286
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,7,0],[0,8,4,1],[1,3,8,5],[1,4,6,6],[2,5,5,9],[2,9,9,8],[3,7,9,4],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,16,10,17],[19,7,20,8],[1,15,2,16],[10,2,11,3],[17,3,18,4],[4,18,5,19],[6,12,7,13],[14,11,15,12],[5,14,6,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(16,1,-17,-2)(18,5,-19,-6)(13,10,-14,-11)(4,11,-5,-12)(12,3,-13,-4)(7,14,-8,-15)(20,15,-9,-16)(2,17,-3,-18)(6,19,-7,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-9)(-2,-18,-6,-20,-16)(-3,12,-5,18)(-4,-12)(-7,-15,20)(-8,9,15)(-10,13,3,17,1)(-11,4,-13)(-14,7,19,5,11)(-17,2)(-19,6)(8,14,10)
Multiloop annotated with half-edges
12^2_302 annotated with half-edges