Configuration spaces of rings and wickets

By Tara E. Brendle, A. Hatcher. 2008

101 citations recorded by the source corpus.

The main result in this paper is that the space of all smooth links in R3 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 aspherical.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ab271485154bde608d7e11bc8e1cdf0a8007d750

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial (2014, background): “The references include Goldsmith [40], McCool [74], Fenn, Rimányi and Rourke [39], Satoh [81], and Brendle and Hatcher [28].”
Representation stability for the cohomology of the pure string motion groups (2011, background): “Recently, Brendle and Hatcher have shown in [3] that it suffices to consider configuration spaces of ‘rings’, that is, embeddings of Euclidean circles in R3 .”
A journey through loop braid groups (2016, methodology): “1We reverse the notations used in [13] for σi and ρi, see also, for example, notations used in [9, 23, 26]”