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arxiv: math/9909095 · v1 · submitted 1999-09-16 · 🧮 math.GT

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Spaces of Knots

Allen Hatcher (Cornell University)
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classification 🧮 math.GT
keywords knotsspherehyperbolicknotsatellitespacesubspaceaction
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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.)

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The Ideal Stratum and Deformation Persistence of Knot Types

    math.GT 2026-04 unverdicted novelty 8.0

    The ropelength of a knot type is the birth level of its admissible deformation components, which carry a new ropelength ultrapseudometric measuring ideal merge scales.

  2. The Ideal Stratum and Deformation Persistence of Knot Types

    math.GT 2026-04 unverdicted novelty 7.0

    Defines ideal strata at ropelength minima for knot types and ideal merge scales that record when distinct ideal components connect under relaxed length bounds, encoded as a zero-dimensional merge persistence via Vieto...