$12^{1}_{337}$ - Minimal pinning sets
Pinning sets for 12^1_337 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_337 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 384
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03466
on average over minimal pinning sets: 2.25
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, 5, 7, 11}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 6, 7, 11}
4
[2, 2, 2, 3]
2.25
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.25
5
0
0
15
2.59
6
0
0
49
2.81
7
0
0
91
2.97
8
0
0
105
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
0
382
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: [[1,2,3,3],[0,4,5,2],[0,1,5,6],[0,6,4,0],[1,3,7,8],[1,8,9,2],[2,7,7,3],[4,6,6,9],[4,9,9,5],[5,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,13,1,14],[14,5,15,6],[6,19,7,20],[12,1,13,2],[4,11,5,12],[15,18,16,19],[7,3,8,2],[8,3,9,4],[17,10,18,11],[16,10,17,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,20,-6,-1)(11,2,-12,-3)(6,13,-7,-14)(14,7,-15,-8)(1,8,-2,-9)(17,10,-18,-11)(12,15,-13,-16)(3,16,-4,-17)(9,18,-10,-19)(19,4,-20,-5)
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,-9,-19,-5)(-2,11,-18,9)(-3,-17,-11)(-4,19,-10,17)(-6,-14,-8,1)(-7,14)(-12,-16,3)(-13,6,20,4,16)(-15,12,2,8)(-20,5)(7,13,15)(10,18)
Loop annotated with half-edges
12^1_337 annotated with half-edges