I will report on recent and ongoing work regarding self-avoiding polygons confined to narrow tubes in the cubic lattice. Two particular problems will be discussed: a classification scheme for "local" and "non-local" knots and their relative frequencies as found with Monte Carlo methods; and a proof of the critical exponent for certain knot types in the $2\times1$ tube.

This is joint work with C. Soteros, J. Eng, K. Shimokawa and K. Ishihara.