Can Non-Smooth C^1 Knots be Orientable?

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The discussion revolves around the orientability of C^1 knots, specifically whether non-smooth embeddings of S^1 into R^3 or S^3 can be guaranteed to be orientable. It is suggested that if S^1 is embedded as a submanifold, it admits tubular neighborhoods, which implies orientability. However, the conversation raises concerns about defining orientability without smoothness, focusing on the role of homology and fundamental classes in this context. The participants explore various definitions of orientable bundles and the implications of homeomorphisms on homology classes. Ultimately, the discussion highlights the complexities of orientability in relation to embeddings and the limitations of certain definitions in manifold theory.
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WWGD said:
Can you find some embedded n-submanifold with non-zero homology in the range {1,2,..,n-1}?

Consider the annulus A = {x in Rn : 1/2 < ||x|| < 1} and note that A is an n-dimensional submanifold of Rn. Then A deformation retracts onto the (n-1)-sphere and therefore has non-zero homology in dimension n-1.
 

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