Operads and knot spaces

By Dev Sinha. 2004

117 citations recorded by the source corpus.

Let Em denote the space of embeddings of the interval I = [?1,1] in the cube Im with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot; see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent to Emb(I,Im) x i?lmm(l, Im). In [28], McClure and Smith define a cosimplicial object O* associated to an operad with multiplication 0, whose homotopy invariant totalization we denote Tot(?#); see Definition 2.17 and Definition 2.5 below. Let /Cm denote the rath Kontsevich operad, introduced in [22], whose entries are compactified configuration spaces and which is weakly equivalent to the little ra-disks operad [37]; see Definition 4.1 and Theorem 4.5.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/109a99fe2c6e2da8a894aaf8c8d7dd969938f42a

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Citations in the corpus

A family of embedding spaces (2006, background): “By comparing with the work of Turchin and Sinha this allows the computation of 2n 3j 3Kn;j .”
On the other side of the bialgebra of chord diagrams (2004, methodology): “ach one ”approximates” the space of knots Emb= K\Σ by means of homotopy limits of diagrams of maps. This approach was initiated by T. Goodwillie and M. Weiss [14, 15], and then developped by D. Sinha [24, 25], and also by I. Volic [37]. In particular for the space of long knots this method provides ”
The rational homotopy of mapping spaces of E${}_n$ operads (2017, methodology): “We use that the operad Assoc is equipped with a filtration (the Hodge filtration) such that Pois = grAssoc (see for instance [50, section 3]).”
Manifold-theoretic compactifications of configuration spaces (2003, methodology): “f P is a polytope each of whose faces (including itself) is homeomorphic to a disk, then the realization of the category of strata of P is diﬀeomorphic to its barycentric subdivision. In further work [23] we plan to show that the spaces Ce n[Rm] form an operad. This construction uniﬁes the associahe”
Hodge‐type decomposition in the homology of long knots (2010, background): “Sinha constructed another cosimplicial model for Embd, which is homotopy equivalent to the one that was briefly described in Section 2.”
Long knots and maps between operads (2010, background): “Starting from work of Goodwillie, Klein, and Weiss [24] [10] [8] [9] on embedding spaces, Sinha [23] has proved that the space of tangentially straightened long knots in Rm is equivalent to the homotopy limit of an explicit cosimplicial space constructed from the m’th Kontsevich operad Km.”
Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots (2011, background): “plex Em,n π computing Q⊗π∗(Embc(Rm,Rn)), see Section 2. This fact is used in Lemma A.1. 1.5. Operads. In recent years it became clear that manifold calculus is deeply related to the theory of operads [35, 3, 4]. This paper continues this tradition. We will now review brieﬂy some relevant notions abo”
The rational homology of spaces of long knots in codimension > 2 (2007, methodology): “Goodwillie-Weiss embedding calculus for knots was developed further by Sinha in [29], who also emphasized the connection with the little d-disks operad in [31].”