$12^{1}_{140}$ - Minimal pinning sets
Pinning sets for 12^1_140 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_140 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 144
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97193
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 4, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 3, 4, 7, 8, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
D (optimal)
• {1, 3, 4, 7, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
4
0
0
2.33
7
0
0
20
2.66
8
0
0
41
2.89
9
0
0
44
3.06
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
140
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,2],[0,1,5,0],[0,6,6,1],[1,7,5,5],[2,4,4,8],[3,8,9,3],[4,9,9,8],[5,7,9,6],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[20,15,1,16],[16,13,17,14],[14,19,15,20],[1,12,2,13],[17,9,18,8],[18,7,19,8],[11,2,12,3],[9,5,10,4],[6,3,7,4],[10,5,11,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,20,-10,-1)(4,1,-5,-2)(2,7,-3,-8)(8,3,-9,-4)(16,5,-17,-6)(6,15,-7,-16)(19,10,-20,-11)(14,11,-15,-12)(18,13,-19,-14)(12,17,-13,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,-9)(-2,-8,-4)(-3,8)(-5,16,-7,2)(-6,-16)(-10,19,13,17,5,1)(-11,14,-19)(-12,-18,-14)(-13,18)(-15,6,-17,12)(-20,9,3,7,15,11)(10,20)
Loop annotated with half-edges
12^1_140 annotated with half-edges