# Spaces of Knots

@article{Hatcher1999SpacesOK, title={Spaces of Knots}, author={Allen Hatcher}, journal={arXiv: Geometric Topology}, year={1999} }

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…

## 19 Citations

### Spaces of knots in the solid torus, knots in the thickened torus, and links in the 3-sphere

- MathematicsGeometriae 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…

### Spaces of knots in the solid torus, knots in the thickened torus, and irreducible links in the 3-sphere

- 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…

### The topology of spaces of knots

- Mathematics
- 2002

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…

### On the homology of the space of knots

- 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…

### Contact homology and one parameter families of Legendrian knots

- Mathematics
- 2005

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…

### Combinatorial cohomology of the space of long knots

- Mathematics
- 2014

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…

### Infinitely many Lagrangian fillings

- MathematicsAnnals 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…

### Loops of Legendrians in Contact 3-Manifolds

- MathematicsSpringer 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…

### A Kontsevich integral of order 1

- MathematicsAlgebraic & Geometric Topology
- 2022

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…

### A family of embedding spaces

- 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…

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