$12^{1}_{331}$ - Minimal pinning sets
Pinning sets for 12^1_331 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_331 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 320
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03463
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 7, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 7, 9, 12}
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
4
1
0
0
2.25
5
0
1
8
2.56
6
0
0
34
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
1
318
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,6,3],[0,2,6,4],[1,3,6,7],[1,7,8,2],[2,8,4,3],[4,9,9,5],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[15,20,16,1],[19,14,20,15],[16,7,17,8],[1,8,2,9],[9,18,10,19],[6,13,7,14],[17,3,18,2],[10,5,11,6],[12,3,13,4],[4,11,5,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,15,-1,-16)(10,1,-11,-2)(8,3,-9,-4)(14,5,-15,-6)(19,6,-20,-7)(2,9,-3,-10)(4,11,-5,-12)(17,12,-18,-13)(7,16,-8,-17)(13,18,-14,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-3,8,16)(-2,-10)(-4,-12,17,-8)(-5,14,18,12)(-6,19,-14)(-7,-17,-13,-19)(-9,2,-11,4)(-15,20,6)(-16,7,-20)(-18,13)(1,15,5,11)(3,9)
Loop annotated with half-edges
12^1_331 annotated with half-edges