$12^{1}_{265}$ - Minimal pinning sets
Pinning sets for 12^1_265 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_265 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 144
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: 2.97944
on average over minimal pinning sets: 2.375
on average over optimal pinning sets: 2.375
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 4, 5, 6, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 4, 6, 8, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
D (optimal)
• {1, 2, 3, 5, 6, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
4
0
0
2.38
7
0
0
20
2.68
8
0
0
41
2.9
9
0
0
44
3.06
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
140
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,7,7,3],[0,2,8,8],[0,8,8,9],[1,9,9,6],[1,5,9,7],[1,6,2,2],[3,4,4,3],[4,6,5,5]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[17,10,18,11],[19,4,20,5],[12,4,13,3],[1,14,2,15],[16,7,17,8],[9,6,10,7],[18,6,19,5],[13,2,14,3],[15,9,16,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(1,10,-2,-11)(17,2,-18,-3)(3,16,-4,-17)(7,4,-8,-5)(14,5,-15,-6)(6,13,-7,-14)(15,8,-16,-9)(19,12,-20,-13)(9,18,-10,-19)
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,-11)(-2,17,-4,7,13,-20,11)(-3,-17)(-5,14,-7)(-6,-14)(-8,15,5)(-9,-19,-13,6,-15)(-10,1,-12,19)(-16,3,-18,9)(2,10,18)(4,16,8)(12,20)
Loop annotated with half-edges
12^1_265 annotated with half-edges