EMBEDDINGS AND IMMERSIONS OF TROPICAL CURVES
Dustin Cartwright, Andrew Dudzik, Madhusudan Manjunath, Yuan Yao · 2016
Abstract. We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Finally, we observe that our immersed tropical curves can be lifted to nodal algebraic curves using the results of Mikhalkin and Shustin. 1.