$11^{1}_{1}$ - Minimal pinning sets

Minimal pinning semi-lattice

(y-axis: cardinality)

Pinning data

• Pinning number of this loop: 9
• Total number of pinning sets: 4
• 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: 2.66818
  • on average over minimal pinning sets: 2.0
  • on average over optimal pinning sets: 2.0

Refined data for the minimal pinning sets

Data for pinning sets in each cardinal

Other information about this loop

Properties

• Region degree sequence: [2, 2, 2, 2, 2, 2, 2, 2, 2, 9, 9]
• Minimal region degree: 2
• Is multisimple: No

Combinatorial encoding data

• Plantri embedding: [[1,1,2,2],[0,3,3,0],[0,4,4,0],[1,5,5,1],[2,6,6,2],[3,7,7,3],[4,8,8,4],[5,8,8,5],[6,7,7,6]]
• PD code (use to draw this loop with SnapPy): [[9,18,10,1],[17,8,18,9],[10,2,11,1],[7,16,8,17],[2,12,3,11],[15,6,16,7],[12,4,13,3],[5,14,6,15],[4,14,5,13]]
• Permutation representation (action on half-edges):
  • Vertex permutation $\sigma=$ (1,10,-2,-11)(13,4,-14,-5)(15,6,-16,-7)(9,18,-10,-1)(11,2,-12,-3)(3,12,-4,-13)(5,14,-6,-15)(7,16,-8,-17)(17,8,-18,-9)
  • 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)
  • Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-11,-3,-13,-5,-15,-7,-17,-9)(-2,11)(-4,13)(-6,15)(-8,17)(-10,1)(-12,3)(-14,5)(-16,7)(-18,9)(2,10,18,8,16,6,14,4,12)

Loop annotated with half-edges