Connected Sums and Decompositions of Plane Curves
Grostic, Christian J.
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Despite recent advances, the problem of plane curve classification remains unsolved. One possible aid in classification would be to examine compositions and decompositions of curves. We use the connected sum action to do just that, examining some of the implications this produces with regard to previous work and introducing some new properties and tools to aid in examination. We do the same for the special case of tree-like curves, relating the connected sum to A-structures and using that as our environment for investigation. We then turn to the question of prime decomposition of plane curves, including the proof that a curve has a unique prime decomposition. Next, we take a look at how index is affected by the connected sum. We conclude with a discussion of the products of the connected sum, with a focus on conditions under which the connected sum is well-defined. To this end, we also discuss symmetry and relate symmetry on tree-like curves to symmetry on A-structures.