Little cubes and long knots

By Ryan Budney. 2003

85 citations recorded by the source corpus.

Abstract This paper gives a partial description of the homotopy type of K , the space of long knots in R 3 . The primary result is the construction of a homotopy equivalence K ≃ C 2 ( P ⊔ { ∗ } ) where C 2 ( P ⊔ { ∗ } ) is the free little 2-cubes object on the pointed space P ⊔ { ∗ } , where P ⊂ K is the subspace of prime knots, and ∗ is a disjoint base-point. In proving the freeness result, a close correspondence is discovered between the Jaco–Shalen–Johannson decomposition of knot complements and the little cubes action on K . Beyond studying long knots in R 3 we show that for any compact manifold M the space of embeddings of R n × M in R n × M with support in I n × M admits an action of the operad of little ( n + 1 ) -cubes. If M = D k this embedding space is the space of framed long n -knots in R n + k , and the action of the little cubes operad is an enrichment of the monoid structure given by the connected-sum operation.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9b47426dd16c038e8a4b5823780de75c843414a4

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Operads and knot spaces (2004, methodology): “brechts establishes a formality result for this model, which determines the rational homotopy type of Em for m > 3. A two-cubes action has been used with spectacular success in dimension three. In [7], Budney constructs a little two-cubes action directly on the space of framed (long) knots in any”
Generalized string topology operations (2006, background): “[2] R. Budney, Little cubes and long knots, preprint: math.”
Hodge‐type decomposition in the homology of long knots (2010, methodology): “Budney constructed a model for the spaces Embframedd , with d 3, that has a natural C2-action [6].”
Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots (2011, background): “It is well-known that the space Embc(R,R), n > 2m+2, is a connected (m + 1)-loop space [6, 40].”
The topology of spaces of knots: cosimplicial models (2002, methodology): “For knots in three dimensions, Budney has used techniques of Hatcher [30] and decomposition theorems for three manifolds to show that this two-cubes action is free [11] and identify the homotopy type of the knot space [12].”