$12^{1}_{205}$ - Minimal pinning sets
Pinning sets for 12^1_205 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_205 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 256
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: 3.0346
on average over minimal pinning sets: 2.4
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 3, 10, 11}
5
[2, 2, 2, 3, 3]
2.40
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
2.4
6
0
0
19
2.68
7
0
0
51
2.89
8
0
0
75
3.03
9
0
0
65
3.15
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
0
253
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,5],[0,5,6,4],[1,3,6,7],[2,8,3,2],[3,9,7,4],[4,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,11,1,12],[12,19,13,20],[13,10,14,11],[1,9,2,8],[18,7,19,8],[9,14,10,15],[2,17,3,18],[3,6,4,7],[15,4,16,5],[5,16,6,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(19,2,-20,-3)(14,3,-15,-4)(4,17,-5,-18)(9,6,-10,-7)(16,7,-17,-8)(5,10,-6,-11)(1,12,-2,-13)(18,13,-19,-14)(8,15,-9,-16)
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,-13,18,-5,-11)(-2,19,13)(-3,14,-19)(-4,-18,-14)(-6,9,15,3,-20,11)(-7,16,-9)(-8,-16)(-10,5,17,7)(-12,1)(-15,8,-17,4)(2,12,20)(6,10)
Loop annotated with half-edges
12^1_205 annotated with half-edges