Has anybody done any work on making a classification of low-complexity tangles, analogous to the work for knots and links? I expect most of the small ones to be rational, and those that aren't rational to be decomposable in the way the square knot is decomposable. But there are more, and I'd like to see a list of the small ones; in particular, the non-mathematician usage of "knot" is often referring to something that could be formalized like this.



Classification of tangles?
As Sam Nead points out, there is a relationship between spatial graphs and tangles obtained by drilling out one edge of a spatial graph having two vertices or drilling out the vertex of a spatial graph having a single vertex. There have been a few people working on building tables of spatial graphs. Here are a few references:


Moriuchi "An enumeration of theta curves with up to 7 crossings" (MR2507922)   

Ishii, et al "A table of genus 2 handlebody knots up to 6 crossings" (MR2890461)   

Lee and Lee "Inequivalent handlebody-knots with homeomorphic complements"(MR2928904)


A handlebody knot is the regular neighborhood of a spatial graph and equivalence of handlebody knots corresponds to neighborhood equivalence of spatial graphs.


