> Every knot is “homeomorphic” to the circle

Here's an explanation:

https://math.stackexchange.com/questions/3791238/introductio...

This isn't true. Knots are not topological spaces, they are imbeddings, K:S^1 -> S^3 (more generally, S^n-> S^n+2). Therefore there isn't an obvious notion of homeomorphism. What the poster points out is that restriction to the image is a homeomorphism from the circle (because it is an imbedding of the circle)

As maps between topological spaces (and almost always we pick these to be from the smooth or piecewise linear categories, which further restricts them), the closest "natural" interpretation of isomorphism is pairs of homeomorphisms, f:S^1 -> S^1, g:S^3 -> S^3 satisfying Kf=gK. Ie natural transformations or commuting squares.

This gives us almost what we want, except that I can flip the orientation of space, or the orientation of the knot with homeomorphisms, neither of which correspond to the physical phenomenon of knots, so we give our spaces an orientation, which requires us to move to the PL or smooth category, or use homotopy/isotopy.

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Koshkin 8 months ago | parent | next [-]

So, in other words, the notion of the homomorphism is almost useless in this context - which is why its mentioning when talking about knots may create confusion.

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gowld 8 months ago | parent | prev [-]

https://en.wikipedia.org/wiki/Embedding#General_topology

"In general topology, an embedding is a homeomorphism onto its image."

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noqc 8 months ago | parent [-]

a homeomorphism to the image is not the same as a homeomorphism.

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fsckboy 7 months ago | parent [-]

onto is not the same as to

	▲	noqc 7 months ago \| parent [-]
		everything is onto its image.

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bmitc 8 months ago | parent | prev | next [-]

Intuitively, just imagine picking a starting point on each of the circle and the knot. Now walk at different speeds such that you get back to the starting point at the same time.

In fact, that's what the knot is: a continuous, bijective mapping from the circle to the image of the mapping, i.e., the knot. (As the linked answer says.)

Edit: I see now that the article already has this intuitive explanation but with ants.

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Koshkin 8 months ago | parent [-]

Somewhat counterintuitively, all knots are homeomorphic to each other.

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lupire 8 months ago | parent | next [-]

Only because "homeomorphic" is a highly technical term that most people don't even know the definition of, let alone have an intuition for, and because "knot" in math is a closed loop, unlikely "knot" in common language.

Once you know the definitions and have learned to tie your shoes, it's quite intuitive. Even a small child can easily constructively prove that knots are homeomorphic.

	▲	bmitc 8 months ago \| parent [-]
		Homeomorphic is also interesting because it's a very fancy sounding word for a concept that has very few requirements to be met. It's usually a basic requirement for concepts in topology, but it does very little to distinguish topological spaces. It's essentially a highfalutin word for "these things are basically the same thing in a very basic way".

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xanderlewis 8 months ago | parent | prev [-]

If you regard two spaces being homeomorphic as meaning — roughly — that if you lived in either space you’d not notice a difference, it makes sense. To a one-dimensional being (that has no concept of curvature or length, since we’re talking about topology here), they’d all feel like living in a circle.

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Nihilartikel 8 months ago | parent | prev [-]

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