$12^{1}_{198}$ - Minimal pinning sets
Pinning sets for 12^1_198 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_198 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 160
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97043
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 7, 9, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {2, 3, 5, 6, 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
5
1
0
0
2.2
6
0
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,3],[0,2,6,6],[0,7,8,8],[1,9,7,1],[2,7,3,3],[4,6,5,9],[4,9,9,4],[5,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,15,9,16],[16,19,17,20],[17,6,18,7],[1,13,2,12],[14,9,15,10],[5,18,6,19],[13,5,14,4],[2,11,3,12],[10,3,11,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,20,-8,-1)(1,10,-2,-11)(14,3,-15,-4)(17,4,-18,-5)(11,6,-12,-7)(19,8,-20,-9)(9,18,-10,-19)(5,12,-6,-13)(2,15,-3,-16)(13,16,-14,-17)
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,-11,-7)(-2,-16,13,-6,11)(-3,14,16)(-4,17,-14)(-5,-13,-17)(-8,19,-10,1)(-9,-19)(-12,5,-18,9,-20,7)(-15,2,10,18,4)(3,15)(6,12)(8,20)
Loop annotated with half-edges
12^1_198 annotated with half-edges