We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> in a compact orientable surface S. The surface S is presented via a triangulation or a handle structure, and the 1-manifolds are given in normal form via their normal coordinates. The running time is bounded above by a polynomial function of the number of triangles in the triangulation (or the number of handles in the handle structure), and the logarithm of the weight of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>. This algorithm represents an improvement over previous work, since its running time depends polynomially on the size of the triangulation of S and it can deal with closed surfaces, unlike many earlier algorithms. Another algorithm, with similar bounds on its running time, can determine whether <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$C_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> are isotopic. We also present a closely related algorithm that can be used to place a standard 1-manifold into normal form.

Some Fast Algorithms for Curves in Surfaces

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2026年4月21日

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Marc Lackenby

摘要

We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds $C_1$ $C_{1}$ and $C_2$ $C_{2}$ in a compact orientable surface S. The surface S is presented via a triangulation or a handle structure, and the 1-manifolds are given in normal form via their normal coordinates. The running time is bounded above by a polynomial function of the number of triangles in the triangulation (or the number of handles in the handle structure), and the logarithm of the weight of $C_1$ $C_{1}$ and $C_2$ $C_{2}$ . This algorithm represents an improvement over previous work, since its running time depends polynomially on the size of the triangulation of S and it can deal with closed surfaces, unlike many earlier algorithms. Another algorithm, with similar bounds on its running time, can determine whether $C_1$ $C_{1}$ and $C_2$ $C_{2}$ are isotopic. We also present a closely related algorithm that can be used to place a standard 1-manifold into normal form.