Measuring complexity of curves on surfaces
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We consider the relations between different measures of complexity for free homotopy classes of curves on a surface \(\Sigma \), including the minimum number of self-intersections, the minimum length of the words representing them in \(\pi _1(\Sigma )\), and the minimum degree of the coverings of \(\Sigma \) to which they lift as embeddings.
KeywordsImmersed curves on surfaces Self-intersections Coverings Word length
Mathematics Subject Classification (2000)Primary 57M05 57M10 Secondary 20F05
The authors would like to thank the referee for helpful comments and corrections. The first author is grateful to the Instituto de Matemáticas, UNAM, for an Undergraduate Fellowship during which this work was completed.