[Paper Review] Spaces of Knots

Allen Hatcher|arXiv (Cornell University)|Sep 16, 1999

Geometric and Algebraic Topology11 references18 citations

TL;DR

This paper investigates the topology of spaces of smooth knots in the 3-sphere isotopic to a given knot, showing that for torus knots and many hyperbolic knots, the space deformation retracts onto the SO(4)-orbit of a maximally symmetric knot placement. The result holds generally for hyperbolic knots if no exotic free actions of finite cyclic groups exist on the 3-sphere.

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.)

Motivation & Objective

• To identify small deformation retract subspaces within the space of all smooth knots isotopic to a given knot in the 3-sphere.
• To determine whether such subspaces can be realized as orbits under the SO(4) action on the 3-sphere.
• To clarify the topological structure of knot spaces, particularly for hyperbolic and satellite knots.
• To establish conditions under which the deformation retraction holds, especially for hyperbolic knots.

Proposed method

• The authors analyze the space of smooth knots isotopic to a fixed knot in the 3-sphere using topological methods.
• They consider the action of SO(4) on the 3-sphere, which corresponds to rotations preserving the ambient geometry.
• For torus and many hyperbolic knots, they show the space deformation retracts onto the SO(4)-orbit of a maximally symmetric knot configuration.
• The argument relies on the assumption that there are no exotic free actions of finite cyclic groups on the 3-sphere, which would otherwise obstruct the result for hyperbolic knots.
• For satellite knots, the structure is more complex but still describable in geometric terms, though full details are omitted in this version.

Experimental results

Research questions

• RQ1 Can the space of all knots isotopic to a given knot deformation retract onto a smaller, symmetric subspace?
• RQ2 Is the SO(4)-orbit of a maximally symmetric knot placement a deformation retract for torus knots?
• RQ3 Under what conditions does the same deformation retract hold for hyperbolic knots?
• RQ4 How does the presence of exotic free actions of finite cyclic groups on the 3-sphere affect the topological structure of knot spaces?
• RQ5 What is the geometric and topological description of the knot space for satellite knots?

Key findings

• For torus knots, the space of isotopic smooth knots deformation retracts onto the SO(4)-orbit of a maximally symmetric embedding.
• For many hyperbolic knots, the same deformation retraction holds, contingent on the nonexistence of exotic free actions of finite cyclic groups on the 3-sphere.
• The deformation retract is realized via the natural SO(4) action on the 3-sphere, which preserves the knot's symmetry.
• The result suggests a strong topological simplification of knot space, reducing it to a homogeneous space under group action.
• For satellite knots, the structure is more complex but still describable in geometric terms, though full analysis is reserved for future work.