$12^{1}_{263}$ - Minimal pinning sets
Pinning sets for 12^1_263 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_263 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 128
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: 2.97249
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 6, 7, 11, 12}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 4, 6, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 3, 4, 6, 7, 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
3
0
0
2.33
7
0
0
16
2.64
8
0
0
35
2.87
9
0
0
40
3.04
10
0
0
25
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
125
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,5,5,0],[0,6,7,7],[1,8,9,1],[1,9,2,2],[3,9,8,7],[3,6,8,3],[4,7,6,9],[4,8,6,5]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,17,9,18],[6,19,7,20],[1,12,2,13],[16,9,17,10],[18,5,19,6],[11,14,12,15],[2,14,3,13],[10,3,11,4],[4,15,5,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(16,3,-17,-4)(18,5,-19,-6)(14,7,-15,-8)(19,10,-20,-11)(11,20,-12,-1)(9,12,-10,-13)(2,13,-3,-14)(6,15,-7,-16)(4,17,-5,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-15,6,-19,-11)(-2,-14,-8)(-3,16,-7,14)(-4,-18,-6,-16)(-5,18)(-9,-13,2)(-10,19,5,17,3,13)(-12,9,1)(-17,4)(-20,11)(7,15)(10,12,20)
Loop annotated with half-edges
12^1_263 annotated with half-edges