An h-principle about Knots

A knot is a smooth map f : S^1 —> R^3 from a circle to 3-space, satisfying two conditions:

1. a global condition: f is injective;
2. a local condition: D_p f is non-singular for each point p in S^1 .

This local condition is an example of of a partial differential relation (PDR), which is like a partial differential equation, but with “=“ replaced by, for instance, “\neq”.  One approach to solving a PDR is to first consider its formal solutions — which are typically easy to understand in the context of homotopy theory — then to ask if such solution is integrable.  

For example, one can ask if 

“D_p f is non-singular for all p” 

by first asking to

“extend f as an injection between tangent bundles”, 

then ask if such an injection between tangent bundles is the derivative of f .  

One says that a PDR satisfies the h-principle if every (compact) family of formal solutions can be, up to homotopy, integrated.  

A celebrated theorem of Gromov, and many others, characterizes a large class of PDRs that satisfy the h-principle.  

Were it not for the global condition of a knot, the space of knots would thusly be without mystery (accepting homotopy theory).  

In this talk, I’ll discuss h-principles, and an h-principle that is compatible with the global condition of knots.  Specifically, I’ll articulate and outline a proof of the following.

Theorem.

The space of framed knots is homotopy equivalent with the space of those knot projections whose writhe vanishes.  

This is an important result for the proof of the tangle hypothesis, which classifies a large class of knot invariants, as well as in the construction of derived Skein modules of 3-manifolds.

This is joint work with John Francis.