• Corpus ID: 17026531

# Spaces of Knots

```@article{Hatcher1999SpacesOK,
title={Spaces of Knots},
author={Allen Hatcher},
journal={arXiv: Geometric Topology},
year={1999}
}```
• A. Hatcher
• Published 16 September 1999
• Mathematics
• arXiv: Geometric Topology
We consider the space of all smooth knots in the 3-sphere isotopic to a given knot, with the aim of finding a small subspace onto which this large space deformation retracts. For torus knots and many hyperbolic knots we show the subspace can be taken to be the orbit of a single maximally symmetric placement of the knot under the action of SO(4) by rotations of the ambient 3-sphere. This would hold for all hyperbolic knots if it were known that there are no exotic free actions of a finite cyclic…
• Mathematics
Geometriae Dedicata
• 2021
We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid
• Mathematics
• 2020
We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid
We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of spaces of maps between configuration
• Mathematics
• 2005
Consider the space of long knots in R , Kn;1 . This is the space of knots as studied by V Vassiliev. Based on previous work (Budney [7], Cohen, Lada and May [12]), it follows that the rational
We consider S 1 -families of Legendrian knots in the standard contact R 3 . We define the monodromy of such a loop, which is an automorphism of the Chekanov-Eliashberg contact homology of the
The motivation of this work is to define cohomology classes in the space of knots that are both easy to find and to evaluate, by reducing the problem to simple linear algebra. We achieve this goal by
• Mathematics
Annals of Mathematics
• 2022
We prove that all maximal-tb Legendrian torus links (n,m) in the standard contact 3-sphere, except for (2,m),(3,3),(3,4) and (3,5), admit infinitely many Lagrangian fillings in the standard
• Mathematics
Springer Proceedings in Physics
• 2019
We study homotopically non-trivial spheres of Legendrians in the standard contact \({\mathbb {R}}^3\) and \({\mathbb {S}}^3\). We prove that there is a homotopy injection of the contactomorphism
We define a 1-cocycle in the space of long knots that is a natural generalisation of the Kontsevich integral seen as a 0-cocycle. It involves a 2-form that generalises the Knizhnik--Zamolodchikov
• Mathematics
• 2007
Let Emb(S, S) denote the space of C∞ -smooth embeddings of the j -sphere in the n-sphere. This paper considers homotopy-theoretic properties of the family of spaces Emb(S , S) for n ≥ j > 0. There is

## References

SHOWING 1-10 OF 16 REFERENCES

• Mathematics
• 1997
The main theorem shows that if M is an irreducible compact connected ori- entable 3{manifold with non-empty boundary, then the classifying space BDi (M rel @M) of the space of dieomorphisms of M
In [1], P. A. Smith has proved that the set of fixed points of a periodic homeomorphism T of the 3-sphere S3 onto itself is an i-sphere imbedded in S3, i = -1, 0, 1, or 2, where the -1-sphere denotes
The Smale Conjecture [9] is the assertion that the inclusion of the orthogonal group 0(4) into Diff(S3), the diffeomorphism group of the 3-sphere with the Cw topology, is a homotopy equivalence.
In this paper the homotopy type of the group of diffeomorphism of sufficiently large irreducible three-dimensional manifolds is described and the space of incompressible surfaces in such manifolds is
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