以结点为界的不可定向曲面：一个地理问题
Nonorientable surfaces bounded by knots: a geography problem
摘要：
不可定向的4-亏格是纽结的一个不变量，许多作者对此进行了研究，包括Gilmer和Livingston、Batson、Ozsváth、Stipsicz和Szabó。给定一个不可定向曲面$F \subset B^4$，其中$\partial F = K\subset S^3$是一个纽结，通过对现有的不可定向曲面4-亏格的定界和计算方法的分析，揭示了$F$的第一Betti数$β_1$与$F$的正规Euler类$e$之间的关系.这种关系产生了一个地理问题：给定一个结$K$，可实现对$(e(F), β_1(F))$的集合是什么，其中$F\subset B^4$是由$K$限定的不可定向曲面？我们探讨这个问题的家庭环面结。此外，我们使用Ozsváth-Szabó $d$ -不变量的双重分支覆盖，以提供更精细的信息的地理问题。我们提出了一个无限家庭的结，这些信息提供了一个改进后的界限Ozsváth，Stipsicz，和Szabó使用Upperstance不变量。
英文摘要：
The nonorientable 4-genus is an invariant of knots which has been studied by many authors, including Gilmer and Livingston, Batson, and Ozsváth, Stipsicz, and Szabó. Given a nonorientable surface $F \subset B^4$ with $\partial F = K\subset S^3$ a knot, an analysis of the existing methods for bounding and computing the nonorientable 4-genus reveals relationships between the first Betti number $β_1$ of $F$ and the normal Euler class $e$ of $F$. This relationship yields a geography problem: given a knot $K$, what is the set of realizable pairs $(e(F), β_1(F))$ where $F\subset B^4$ is a nonorientable surface bounded by $K$? We explore this problem for families of torus knots. In addition, we use the Ozsváth-Szabó $d$-invariant of two-fold branched covers to give finer information on the geography problem. We present an infinite family of knots where this information provides an improvement upon the bound given by Ozsváth, Stipsicz, and Szabó using the Upsilon invariant.
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