$12^{1}_{232}$ - Minimal pinning sets
Pinning sets for 12^1_232 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_232 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 96
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: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 6, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 4, 6, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,3,3],[0,2,2,6],[0,7,8,8],[1,9,9,1],[3,9,7,7],[4,6,6,8],[4,7,9,4],[5,8,6,5]]
PD code (use to draw this loop with SnapPy): [[17,20,18,1],[5,16,6,17],[6,19,7,20],[18,7,19,8],[1,10,2,11],[15,4,16,5],[8,14,9,13],[9,12,10,13],[2,12,3,11],[3,14,4,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,2,-14,-3)(16,5,-17,-6)(6,17,-7,-18)(4,7,-5,-8)(8,3,-9,-4)(18,9,-19,-10)(20,11,-1,-12)(1,14,-2,-15)(12,15,-13,-16)(10,19,-11,-20)
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,-15,12)(-2,13,15)(-3,8,-5,16,-13)(-4,-8)(-6,-18,-10,-20,-12,-16)(-7,4,-9,18)(-11,20)(-14,1,11,19,9,3)(-17,6)(-19,10)(2,14)(5,7,17)
Loop annotated with half-edges
12^1_232 annotated with half-edges