Complementary regions for immersions of surfaces

2007/01/07 by Tahl Nowik, Nowik, Tahl
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Geometric Topology (math.GT) #math.GT

paper · pdf · doi:10.48550/arxiv.math/0701208

arxiv created 2007/01/07 · openalex publication_date 2007/01/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F be a closed surface and i:F → S³ a generic immersions. Then S³ - i(F) is a union of connected regions, which may be separated into two sets U_j and V_j by a checkerboard coloring. For k ≥ 0, let a_k, b_k be the number of components U_j, V_j with χ(U_j) = 1-k, χ(V_j)=1-k, respectively. Two more integers attached to i are the number N of triple points of i, and χ=χ(F). In this work we determine what sets of data (a_k, b_k, χ, N) may appear in this way.

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