It is a classical result that Newton's method converges rapidly to a nondegenerate zero if the initial condition is sufficiently close to the zero. In a recent work, Dupont and Scott ask the following question: do iterates converge to the limit in a particular direction? In the case of a planar system, their numerical simulations suggest that this is often but not always the case. Since, understanding finer information about the asymptotic dynamics may have implications for further algorithm refinement, have a more precise quantitative answer to this question, both in a positive and negative sense seems desirable. The challenge is that a reasonable model for the asymptotic dynamics consists of a multi parameter family of nonlinear circle maps, where the parameter space is a solid 4-dimensional tours. I will describe our efforts to use a combinatorial, algebraic topological description of the global dynamics to provide a description of the possible dynamics over all of parameter space. This is joint work with J. Bush, W. Cowen, and S. Harker