Spaces of Knots
arXiv:math/9909095
Abstract
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 group on the 3-sphere. For satellite knots the situation is more complicated but still describable in fairly simple terms. (This preliminary version of the paper does not include details for the case of satellite knots.)
Preliminary version
Cited by in corpus (8)
- Contact homology and one parameter families of Legendrian knots
- The topology of spaces of knots
- Infinitely many Lagrangian fillings
- The Legendrian Whitney trick
- Combinatorial cohomology of the space of long knots
- Fundamental groups of formal Legendrian and horizontal embedding spaces
- Finite-type 1-cocycles of knots given by Polyak-Viro formulas
- A Kontsevich integral of order 1