  We give a complete classification of homomorphisms from the braid group on
$n$ strands to the braid group on $2n$ strands when $n$ is at least 5. We also
classify endomorphisms of the braid group on 4 strands, as well as
homomorphisms from the commutator subgroup of the braid group on $n$ strands to
the braid group on $2n-5$ strands. Our classifications suggest a recursive
classification of homomorphisms between any braid groups. We also give a
simple, geometric proof of a theorem of Lin that highly constrains the
holomorphic maps that may exist between spaces of monic, square-free
polynomials of two given degrees.
