论文标题
不可定向的表面以结的界定：地理问题
Nonorientable surfaces bounded by knots: a geography problem
论文作者
论文摘要
不可定向的4摄氏度是结的不变，许多作者都研究了，包括吉尔默和利文斯顿，巴特森和奥兹瓦斯，斯蒂普西兹和szabó。给定一个不可定向的表面$ f \子集b^4 $，带有$ \ partial f = k \ subset s^3 $一个结，对现有的界限和计算现有方法的分析揭示了第一个betti数字$β_1$ f $ $ f $ $β_1$ f $与普通的Euler euler类$ e $ f $ $ f $。这种关系产生了一个地理问题：给定一个结$ k $，一组可实现的对$（e（f），β_1（f））$，其中$ f \ subset b^4 $是由$ k $限制的不可定向的表面？我们为Torus结的家庭探索了这个问题。此外，我们使用Ozsváth-Szabó$ d $ d $ invariant的两倍分支封面，以提供有关地理问题的更好信息。我们提出了一个无限的结家族，其中此信息可以使用Upsilon不变式对Ozsváth，Stipsicz和Szabó的限制进行改进。
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.
