$12^{1}_{320}$ - Minimal pinning sets
Pinning sets for 12^1_320 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_320 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 512
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: 3.10608
on average over minimal pinning sets: 2.5
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 5, 7, 11}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 5, 8, 11}
4
[2, 2, 3, 3]
2.50
C (optimal)
• {1, 4, 8, 11}
4
[2, 2, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
3
0
0
2.5
5
0
0
22
2.76
6
0
0
70
2.94
7
0
0
126
3.06
8
0
0
140
3.15
9
0
0
98
3.22
10
0
0
42
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
3
0
509
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,3],[0,2,7,8],[0,9,5,5],[1,4,4,6],[1,5,9,2],[3,9,8,8],[3,7,7,9],[4,8,7,6]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,3,11,4],[4,19,5,20],[5,8,6,9],[1,16,2,17],[17,2,18,3],[11,18,12,19],[7,14,8,15],[6,14,7,13],[15,12,16,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(8,1,-9,-2)(3,6,-4,-7)(18,7,-19,-8)(13,10,-14,-11)(11,4,-12,-5)(5,12,-6,-13)(19,14,-20,-15)(9,16,-10,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-19,-15)(-2,-18,-8)(-3,-7,18)(-4,11,-14,19,7)(-5,-13,-11)(-6,3,17,-10,13)(-9,-17,2)(-12,5)(-16,9,1)(-20,15)(4,6,12)(10,16,20,14)
Loop annotated with half-edges
12^1_320 annotated with half-edges