Thank you. Actually random graphs would wotk for me too, I just need a model with variable numbers of nodes and edges at the same time.

Or in other words: what about the vertices that are not in the path?

@DavidE.Roberson I'm not sure I understand your comment. Yes a homomorphic image of a path can be anything but being inside a cube puts further restrictions on the image. For example you can map a spanning path in a 3-cube to a heptagon but the adjacency relations imposed by the cube and the definition of a graph map imply that the image cannot be longer than a 4-cycle.

Yes, sure, braids are tangles. But there are tangles which are not braids namely those poor little caps which have $n$ incoming and $n-2$ outgoing endpoints. For those one needs to define a partial trace on the algebra.

The definition of Jones polynomial from TL algebras uses a trace on the algebra which is given by closing up the element and counting the number of resulting circles. One may define a "partial trace" by closing up only two connected components of this tangle to get the homomorphism I alluded to in my question. I should verify if this gives the right answer.

Thank you, I didn't know about the sign problem. But this still doesn't answer my question. As far as I know the original definition f the Jones polynomial was given in terms of a braid group representation on Hecke algebras so it didn't have much to do with tangles.