$12^{1}_{257}$ - Minimal pinning sets
Pinning sets for 12^1_257 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_257 Pinning data
Pinning number of this loop: 7
Total number of pinning sets: 64
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: 2.9179
on average over minimal pinning sets: 2.28571
on average over optimal pinning sets: 2.28571
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 7, 8, 11}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
B (optimal)
• {1, 2, 3, 4, 7, 9, 11}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
C (optimal)
• {1, 2, 3, 4, 7, 9, 12}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
7
3
0
0
2.29
8
0
0
13
2.63
9
0
0
22
2.9
10
0
0
18
3.1
11
0
0
7
3.25
12
0
0
1
3.33
Total
3
0
61
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 6, 8]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,5,6],[1,7,7,5],[2,4,6,3],[3,5,8,8],[4,9,9,4],[6,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[19,10,20,11],[12,10,13,9],[1,9,2,8],[18,3,19,4],[13,3,14,2],[14,7,15,8],[4,17,5,18],[6,15,7,16],[16,5,17,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,11,-1,-12)(16,1,-17,-2)(14,3,-15,-4)(12,5,-13,-6)(6,19,-7,-20)(7,10,-8,-11)(17,8,-18,-9)(4,13,-5,-14)(2,15,-3,-16)(9,18,-10,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-3,14,-5,12)(-2,-16)(-4,-14)(-6,-20,-12)(-7,-11,20)(-8,17,1,11)(-9,-19,6,-13,4,-15,2,-17)(-10,7,19)(-18,9)(3,15)(5,13)(8,10,18)
Loop annotated with half-edges
12^1_257 annotated with half-edges