$12^{1}_{231}$ - Minimal pinning sets
Pinning sets for 12^1_231 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_231 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 128
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90623
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 11}
5
[2, 2, 2, 2, 2]
2.00
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
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 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],[0,5,6,7],[1,8,8,9],[2,9,3,2],[3,9,9,7],[3,6,8,8],[4,7,7,4],[4,6,6,5]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[19,6,20,7],[8,6,9,5],[1,10,2,11],[13,18,14,19],[9,4,10,5],[2,15,3,16],[11,16,12,17],[17,12,18,13],[14,3,15,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,20,-12,-1)(6,1,-7,-2)(2,5,-3,-6)(16,3,-17,-4)(18,7,-19,-8)(14,9,-15,-10)(19,12,-20,-13)(8,13,-9,-14)(10,15,-11,-16)(4,17,-5,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-3,16,-11)(-2,-6)(-4,-18,-8,-14,-10,-16)(-5,2,-7,18)(-9,14)(-12,19,7,1)(-13,8,-19)(-15,10)(-17,4)(-20,11,15,9,13)(3,5,17)(12,20)
Loop annotated with half-edges
12^1_231 annotated with half-edges