Fibonacci representations of braid groups

A family of representations is constructed for the braid group $B_n$. The vector spaces on which the braid group acts are defined as the result of identifying the spaces generated by proper colorings of regular trees of degree $3$ with a marked vertex. This identification is done using a family of canonical isomorphisms. The dimensions of the resulting spaces form the sequence of Fibonacci numbers. We then show how the constructed representations can be extended to invariants of unoriented knots and links in a 3-sphere.