$12^{1}_{236}$ - Minimal pinning sets
Pinning sets for 12^1_236 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_236 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 224
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: 2.9785
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6, 11}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 5, 11}
5
[2, 2, 2, 2, 4]
2.40
C (optimal)
• {1, 2, 3, 4, 11}
5
[2, 2, 2, 2, 3]
2.20
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
2.27
6
0
0
18
2.59
7
0
0
46
2.82
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
3
0
221
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,3,3],[0,2,2,7],[0,7,8,5],[1,4,6,1],[1,5,8,2],[3,9,9,4],[4,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[17,20,18,1],[9,16,10,17],[19,6,20,7],[18,6,19,5],[1,14,2,15],[15,8,16,9],[10,8,11,7],[13,4,14,5],[2,12,3,11],[3,12,4,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(2,15,-3,-16)(16,3,-17,-4)(4,7,-5,-8)(12,5,-13,-6)(8,19,-9,-20)(20,9,-1,-10)(18,11,-19,-12)(6,13,-7,-14)(14,17,-15,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,-16,-4,-8,-20,-10)(-3,16)(-5,12,-19,8)(-6,-14,-18,-12)(-7,4,-17,14)(-9,20)(-11,18,-15,2)(-13,6)(1,9,19,11)(3,15,17)(5,7,13)
Loop annotated with half-edges
12^1_236 annotated with half-edges