The Tangle Hypothesis, and an h-principle for spaces of knots

The Tangle Hypothesis asserts a universal property for spaces of “framed” 1-dimensional submanifolds of R^n.  For n=3, this is a space of “framed” knots.  Here, “framing” is data concerning tangent bundles.  

 

I will state the Tangle Hypothesis, and outline a proof of it, which uses induction on n.  The inductive step requires a comparison between a “framing” of a 1-dimensional submanifold W of R^n with a more geometric condition: the projection of W onto R^{n-1} is an immersion.  h-principles are designed to solve this sort of comparison.  However, all known proofs of an h-principle apply to sheaves (ie, presheaves whose global values are determined by their local values), while spaces of embeddings Emb(-,R^n) are quintessentially not sheaves.

 

I hope to use the majority of the talk discussing h-principles.  I hope to explain an alternative proof of a large class of h-principles, which can be applied to spaces of embeddings, thereby proving the Tangle Hypothesis.