What Sets Homeomorphic and Isotopic Knots Apart?

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Discussion Overview

The discussion centers on the differences between homeomorphic and isotopic knots, exploring the definitions and implications of these concepts within the context of knot theory.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants clarify that an isotopy involves a smooth path of embeddings between two manifolds, while a homeomorphism is defined as a single function between two manifolds.
  • One participant illustrates the concept of isotopy using the example of a right circular cylinder representing an isotopy between two circles.
  • Another participant notes that while the unlink of two components is homeomorphic to the Hopf link, they are not isotopic.
  • It is proposed that if two knot projections can be deformed into each other through a sequence of Reidemeister moves, then those projections are isotopic.
  • One participant agrees that each Reidemeister move results in an isotopic projection of a knot relative to the original.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of homeomorphism and isotopy, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Some definitions may depend on specific mathematical contexts, and the implications of Reidemeister moves are not fully explored in terms of their limitations or assumptions.

ComputerGeek
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What is the difference?
 
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ComputerGeek said:
What is the difference?
Don't you mean "homeomorphic vs isotropic"?
The root words mean "same form" and "same change" (same difference?). Why not look them up in a scientific dictionary or on Google?
(I looked them up. Never mind, sorry!)
 
Last edited:
An isotopy is a smooth path of embeddings between two manifolds, while a homeomorphism is just a single function between two manifolds. Ie., a right circular cylinder centered at the origin with unit radius is a representation of an isotopy between the two circles at either end.
While the unlink of 2 components is homeomorphic to the Hopf link, the two are not isotopic.
 
Last edited:
hypermorphism said:
An isotopy is a smooth path of embeddings between two manifolds, while a homeomorphism is just a single function between two manifolds. Ie., a right circular cylinder centered at the origin with unit radius is a representation of an isotopy between the two circles at either end.
While the unlink of 2 components is homeomorphic to the Hopf link, the two are not isotopic.
So, it is appropriate to say:

If two knot projections can be deformed into each other via a sequence of Reidemeister moves then the knot projections are isotopic to one another.
 
Yep. Each Reidemeister move produces an isotopic projection of a knot with respect to the original.
 

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