Configuration spaces of rings and wickets

2008/05/28 by Tara Brendle, Brendle, Tara, Allen Hatcher +1 · 4 citations
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:57M25

paper · pdf · doi:10.48550/arxiv.0805.4354

28 pages. Some revisions in the exposition

arxiv created 2010/01/11 · arxiv updated 2010/01/13

Abstract

The main result in this paper is that the space of all smooth links in Euclidean 3-space isotopic to the trivial link of n components has the same homotopy type as its finite-dimensional subspace consisting of configurations of n unlinked Euclidean circles (the "rings" in the title). There is also an analogous result for spaces of arcs in upper half-space, with circles replaced by semicircles (the "wickets" in the title). A key part of the proofs is a procedure for greatly reducing the complexity of tangled configurations of rings and wickets. This leads to simple methods for computing presentations for the fundamental groups of these spaces of rings and wickets as well as various interesting subspaces. The wicket spaces are also shown to be K(G,1)'s.

Cited by

• A presentation for the pure Hilden group
• A basis and Schur-Weyl duality for the loop Hecke algebra
• Dense and empty BNSR-invariants of the McCool groups
• The embedding space of a Hopf link

Related

• Spaces of knots in the solid torus, knots in the thickened torus, and\n links in the 3-sphere
• Configuration spaces of circles in the plane
• Configuration space integrals and the topology of knot and link spaces
• The rack space
• Semiconfiguration Spaces of Planar Linkages