$12^{2}_{277}$ - Minimal pinning sets
Pinning sets for 12^2_277 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_277 Pinning data
Pinning number of this multiloop: 7
Total number of pinning sets: 56
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.86825
on average over minimal pinning sets: 2.19048
on average over optimal pinning sets: 2.19048
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 6, 7, 8, 9, 11}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
B (optimal)
• {1, 2, 4, 7, 8, 9, 11}
7
[2, 2, 2, 2, 2, 2, 4]
2.29
C (optimal)
• {1, 2, 3, 7, 8, 9, 11}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
7
3
0
0
2.19
8
0
0
12
2.58
9
0
0
19
2.87
10
0
0
15
3.07
11
0
0
6
3.21
12
0
0
1
3.33
Total
3
0
53
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 6, 8]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,5,6,3],[0,2,7,7],[1,8,8,1],[2,8,8,9],[2,9,9,7],[3,6,9,3],[4,5,5,4],[5,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,11,14,12],[19,7,20,8],[1,7,2,6],[10,14,11,15],[8,16,9,17],[4,18,5,19],[2,5,3,6],[15,9,16,10],[17,3,18,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,12,-14,-1)(11,2,-12,-3)(3,10,-4,-11)(15,4,-16,-5)(19,6,-20,-7)(17,8,-18,-9)(7,18,-8,-19)(5,20,-6,-13)(1,14,-2,-15)(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,-15,-5,-13)(-2,11,-4,15)(-3,-11)(-6,19,-8,17,-10,3,-12,13)(-7,-19)(-9,-17)(-14,1)(-16,9,-18,7,-20,5)(2,14,12)(4,10,16)(6,20)(8,18)
Multiloop annotated with half-edges
12^2_277 annotated with half-edges