$12^{1}_{304}$ - Minimal pinning sets
Pinning sets for 12^1_304 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_304 Pinning data
Pinning number of this loop: 7
Total number of pinning sets: 48
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.86152
on average over minimal pinning sets: 2.14286
on average over optimal pinning sets: 2.14286
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 6, 7, 11}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
B (optimal)
• {1, 2, 3, 4, 5, 7, 11}
7
[2, 2, 2, 2, 2, 2, 3]
2.14
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
7
2
0
0
2.14
8
0
0
9
2.53
9
0
0
16
2.82
10
0
0
14
3.04
11
0
0
6
3.21
12
0
0
1
3.33
Total
2
0
46
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,5,6,0],[1,6,2,1],[2,7,7,3],[3,8,8,4],[5,9,9,5],[6,9,9,6],[7,8,8,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[8,17,9,18],[19,16,20,17],[10,2,11,1],[18,7,19,8],[15,2,16,3],[11,6,12,7],[3,14,4,15],[5,12,6,13],[13,4,14,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(8,3,-9,-4)(15,4,-16,-5)(2,9,-3,-10)(20,11,-1,-12)(16,13,-17,-14)(5,14,-6,-15)(6,17,-7,-18)(18,7,-19,-8)(12,19,-13,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-3,8,-19,12)(-2,-10)(-4,15,-6,-18,-8)(-5,-15)(-7,18)(-9,2,-11,20,-13,16,4)(-12,-20)(-14,5,-16)(-17,6,14)(1,11)(3,9)(7,17,13,19)
Loop annotated with half-edges
12^1_304 annotated with half-edges