$12^{1}_{234}$ - Minimal pinning sets
Pinning sets for 12^1_234 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_234 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 112
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: 2.91983
on average over minimal pinning sets: 2.22222
on average over optimal pinning sets: 2.22222
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 5, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 3, 4, 5, 10, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
C (optimal)
• {1, 3, 4, 5, 9, 11}
6
[2, 2, 2, 2, 2, 4]
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
2.22
7
0
0
15
2.57
8
0
0
31
2.82
9
0
0
34
3.01
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
3
0
109
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,6,6,0],[0,7,8,4],[1,3,5,1],[1,4,8,9],[2,9,9,2],[3,9,8,8],[3,7,7,5],[5,7,6,6]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[13,6,14,7],[19,8,20,9],[1,4,2,5],[5,12,6,13],[14,12,15,11],[9,18,10,19],[3,16,4,17],[2,16,3,15],[17,10,18,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-19,-2)(10,3,-11,-4)(16,5,-17,-6)(14,7,-15,-8)(8,11,-9,-12)(2,9,-3,-10)(12,19,-13,-20)(20,13,-1,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-5,16,-7,14)(-2,-10,-4,-18)(-3,10)(-6,-16)(-8,-12,-20,-14)(-9,2,-19,12)(-11,8,-15,6,-17,4)(-13,20)(1,13,19)(3,9,11)(5,17)(7,15)
Loop annotated with half-edges
12^1_234 annotated with half-edges