$12^{2}_{263}$ - Minimal pinning sets
Pinning sets for 12^2_263 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_263 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 88
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: 2.91951
on average over minimal pinning sets: 2.29365
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 6, 7}
6
[2, 2, 2, 2, 2, 3]
2.17
a (minimal)
• {1, 2, 3, 4, 5, 7, 10}
7
[2, 2, 2, 2, 2, 3, 4]
2.43
b (minimal)
• {1, 2, 3, 4, 5, 7, 9}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
1
0
0
2.17
7
0
2
6
2.46
8
0
0
22
2.74
9
0
0
29
2.97
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
2
85
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 8]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,6],[1,7,7,5],[2,4,8,6],[3,5,8,3],[4,9,9,4],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,7,10,8],[10,19,11,20],[1,11,2,12],[6,14,7,15],[18,13,19,14],[2,13,3,12],[15,5,16,6],[17,3,18,4],[4,16,5,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(15,2,-16,-3)(11,6,-12,-7)(4,17,-5,-18)(18,5,-19,-6)(12,19,-13,-20)(1,20,-2,-9)(7,10,-8,-11)(16,13,-17,-14)(3,14,-4,-15)
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,-9)(-2,15,-4,-18,-6,11,-8,9)(-3,-15)(-5,18)(-7,-11)(-10,7,-12,-20,1)(-13,16,2,20)(-14,3,-16)(-17,4,14)(-19,12,6)(5,17,13,19)(8,10)
Multiloop annotated with half-edges
12^2_263 annotated with half-edges