$12^{1}_{174}$ - Minimal pinning sets
Pinning sets for 12^1_174 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_174 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 288
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03466
on average over minimal pinning sets: 2.4
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, 4, 5, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 4, 5, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 3, 4, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.4
6
0
0
24
2.69
7
0
0
61
2.9
8
0
0
85
3.05
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
0
284
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,7],[0,7,6,8],[0,8,9,9],[1,9,8,1],[1,3,7,2],[2,6,3,2],[3,5,9,4],[4,8,5,4]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[4,11,5,12],[8,19,9,20],[6,18,7,17],[1,14,2,15],[12,3,13,4],[7,10,8,11],[18,9,19,10],[13,16,14,17],[2,16,3,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(15,6,-16,-7)(4,7,-5,-8)(1,8,-2,-9)(18,11,-19,-12)(12,19,-13,-20)(10,13,-11,-14)(14,3,-15,-4)(5,16,-6,-17)(2,17,-3,-18)
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,-18,-12,-20,9)(-3,14,-11,18)(-4,-8,1,-10,-14)(-5,-17,2,8)(-6,15,3,17)(-7,4,-15)(-13,10,20)(-16,5,7)(-19,12)(6,16)(11,13,19)
Loop annotated with half-edges
12^1_174 annotated with half-edges