What Sets Homeomorphic and Isotopic Knots Apart?

[ComputerGeek]

What is the difference?

 

[MetaKron]

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!)

 

[hypermorphism]

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.

 

[ComputerGeek]

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.

 

[hypermorphism]

Yep. Each Reidemeister move produces an isotopic projection of a knot with respect to the original.