Here's an idea for a knot-based Diffie-Hellman exchange:

- Public:  random (oriented) knot $P$.
- Private:  random (oriented) knots $A$ and $B$.
- Exchange:  Alice sends (randomized or canonical representation of) $A'=A\oplus P$, Bob sends (randomized or canonical representation of) $B'=P\oplus B$.
- Shared key:  (invariant of) $A\oplus P\oplus B=A\oplus B'=A'\oplus B$.

Questions:

- Why is this a good/bad idea?  References?
- What is the complexity of solving $P\oplus X=X'$ for $X$ given $P$ and $X'$?  Is factoring knots difficult?
- What are good digital representations of knots?  How to efficiently generate and randomize knot representations?  (E.g., Gauss codes and Reidemeister moves?)  "Diffusing" the sums $A'$, $B'$ (randomly or to some canonical form) would be important for security, assuming factoring is hard in the first place.
- What is a good choice of invariant in the last step?  Should be efficient to compute for Alice and Bob but infeasible knowing $P$, $A'$, and $B'$.
- If this makes sense, can it be done in reasonable space/time (e.g. $n\log n$ diagram storage for $n$ crossings, obfuscation/diffusion, final invariant computation)?

TL;DR:  Could the knot monoid be useful for public key cryptography?  Is connected sum of knots one-way, even knowing one of the factors?  Do random planar projections obfuscate knot factorizations?