  Let $P$ be a polynomial with a connected Julia set $J$. We use continuum
theory to show that it admits a \emph{finest monotone map $\ph$ onto a locally
connected continuum $J_{\sim_P}$}, i.e. a monotone map $\ph:J\to J_{\sim_P}$
such that for any other monotone map $\psi:J\to J'$ there exists a monotone map
$h$ with $\psi=h\circ \ph$. Then we extend $\ph$ onto the complex plane $\C$
(keeping the same notation) and show that $\ph$ monotonically semiconjugates
$P|_{\C}$ to a \emph{topological polynomial $g:\C\to \C$}. If $P$ does not have
Siegel or Cremer periodic points this gives an alternative proof of Kiwi's
fundamental results on locally connected models of dynamics on the Julia sets,
but the results hold for all polynomials with connected Julia sets. We also
give a criterion and a useful sufficient condition for the map $\ph$ not to
collapse $J$ into a point.
