$12^{2}_{304}$ - Minimal pinning sets
Pinning sets for 12^2_304 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_304 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03441
on average over minimal pinning sets: 2.46667
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 12}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 5, 7, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 4, 5, 7, 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
5
1
0
0
2.4
6
0
2
7
2.63
7
0
0
31
2.83
8
0
0
55
2.99
9
0
0
55
3.12
10
0
0
31
3.22
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
2
189
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 6, 6]
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],[0,8,5,1],[1,4,8,2],[2,7,3,3],[3,6,9,9],[4,9,9,5],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,16,6,17],[17,3,18,4],[19,10,20,11],[1,15,2,16],[6,2,7,3],[18,12,19,11],[12,9,13,10],[14,7,15,8],[8,13,9,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(16,1,-17,-2)(13,6,-14,-7)(11,8,-12,-9)(18,9,-19,-10)(10,17,-11,-18)(7,12,-8,-13)(3,14,-4,-15)(20,15,-5,-16)(2,19,-3,-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,-5)(-2,-20,-16)(-3,-15,20)(-4,5,15)(-6,13,-8,11,17,1)(-7,-13)(-9,18,-11)(-10,-18)(-12,7,-14,3,19,9)(-17,10,-19,2)(4,14,6)(8,12)
Multiloop annotated with half-edges
12^2_304 annotated with half-edges