$12^{1}_{335}$ - Minimal pinning sets
Pinning sets for 12^1_335 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_335 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 256
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: 3.0346
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, 3, 6, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 5, 7, 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
3
0
0
2.4
6
0
0
19
2.68
7
0
0
51
2.89
8
0
0
75
3.03
9
0
0
65
3.15
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
0
253
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,2],[0,1,5,6],[0,7,4,0],[1,3,8,5],[1,4,8,2],[2,9,7,7],[3,6,6,9],[4,9,9,5],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[15,20,16,1],[14,5,15,6],[19,4,20,5],[16,2,17,1],[6,17,7,18],[18,13,19,14],[3,10,4,11],[2,10,3,9],[7,12,8,13],[11,8,12,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(11,2,-12,-3)(8,5,-9,-6)(6,19,-7,-20)(20,7,-1,-8)(4,9,-5,-10)(15,10,-16,-11)(3,14,-4,-15)(12,17,-13,-18)(18,13,-19,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,10,-5,8)(-2,11,-16)(-3,-15,-11)(-4,-10,15)(-6,-20,-8)(-7,20)(-9,4,14,-19,6)(-12,-18,-14,3)(-13,18)(-17,12,2)(1,7,19,13,17)(5,9)
Loop annotated with half-edges
12^1_335 annotated with half-edges