$12^{1}_{2}$ - Minimal pinning sets

Minimal pinning semi-lattice

(y-axis: cardinality)

Pinning data

• Pinning number of this loop: 8
• Total number of pinning sets: 16
• 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: 2.76717
  • 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, 5, 5, 7, 7]
• Minimal region degree: 2
• Is multisimple: No

Combinatorial encoding data

• Plantri embedding: [[1,2,3,3],[0,4,5,5],[0,6,6,4],[0,7,7,0],[1,7,7,2],[1,8,8,1],[2,9,9,2],[3,4,4,3],[5,9,9,5],[6,8,8,6]]
• PD code (use to draw this loop with SnapPy): [[20,13,1,14],[14,10,15,9],[19,4,20,5],[12,1,13,2],[10,3,11,4],[15,8,16,9],[5,18,6,19],[2,11,3,12],[7,16,8,17],[17,6,18,7]]
• Permutation representation (action on half-edges):
  • Vertex permutation $\sigma=$ (7,20,-8,-1)(16,1,-17,-2)(14,3,-15,-4)(12,5,-13,-6)(6,11,-7,-12)(19,8,-20,-9)(17,10,-18,-11)(4,13,-5,-14)(2,15,-3,-16)(9,18,-10,-19)
  • 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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-3,14,-5,12,-7)(-2,-16)(-4,-14)(-6,-12)(-8,19,-10,17,1)(-9,-19)(-11,6,-13,4,-15,2,-17)(-18,9,-20,7,11)(3,15)(5,13)(8,20)(10,18)

Loop annotated with half-edges