Added some pictures and made more changes

Author: Brad-A-M

Min_homotopy.tex

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 An introduction to the minimum homotopy problem is given in \cite{fkw2017}. In section 2 of ~\cite{nie2014} the weighted cancellation norm on the free group is defined. Theorem 4.1 of~\cite{nie2014} proves that the minimum area of a homotopy between two curves $\gamma_1$ and $\gamma_2$ exists, and is equal to the weighted cancellation distance of the words generated by the curves. If we let one of the curves be a point we have solution to the minimum null homotopy problem.