A hyperbolic approach to exp₃(S¹)

2007/08/15 by S. C. F. Rose, Rose, S. C. F.
Mathematics · #54B20 #55Q52 #57M25 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #math.AT #math.GT #msc:54B20 #msc:55Q52 #msc:57M25

paper · pdf · doi:10.48550/arxiv.0708.2085

arxiv created 2007/08/15 · arxiv updated 2009/12/01

Abstract

In this paper we investigate a new geometric method of studying exp_k(S¹), the set of all non-empty subsets of the circle of cardinality at most k. By considering the circle as the boundary of the hyperbolic plane we are able to use its group of isometries to determine explicitely the structure of its first few configuration spaces. We then study how these configuration spaces fit together in their union, exp₃(S¹), to reprove an old theorem of Bott as well as to offer a new proof (following that of E. Shchepin) of the fact that the embedding exp₁(S¹) into exp₃(S¹) is the trefoil knot.

Related

• Lattices in R² and finite subsets of a circle
• The third symmetric potency of the circle and the Barnette sphere
• Configuration spaces of points on the circle and hyperbolic Dehn fillings
• Hyperbolic Knot Theory
• Hyperbolic Knots