发表机构
Smith College（史密斯学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究 Murasugi 和的几何本质性，证明几何本质性不保持，并给出管道无结环带产生可压缩曲面的充分条件，进而构造性证明 Hatcher-Thurston 关于 2-桥纽结的结果。
AI 中文摘要
Gabai 证明了任何 $\u03c0_1$-本质 Seifert 曲面的 Murasugi 和也是 $\u03c0_1$-本质的，Ozawa 将此结果推广到无向跨越曲面。然而，我们表明关于几何本质曲面的类似陈述是不成立的。（如果跨越曲面不能被压缩或边界压缩到另一个跨越曲面，则它是几何本质的。）我们还询问何时将无结环带（带有任意数量的扭转）管道连接到可压缩跨越曲面上会产生可压缩曲面。我们提供了正例和反例，并建立了一个简单的充分条件。作为应用，我们获得了 Hatcher 和 Thurston 关于 2-桥纽结和链环的本质跨越曲面的结果的一个新的构造性证明。
英文摘要
Gabai proved that any Murasugi sum of $π_1$-essential Seifert surfaces is also $π_1$-essential, and Ozawa extended this result to unoriented spanning surfaces. We show, however, that the analogous statement about geometrically essential surfaces is untrue. (A spanning surface is geometrically essential if it cannot be compressed or boundary compressed to another spanning surface.) We also ask when plumbing an unknotted annulus (with any number of twists) onto a compressible spanning surface yields a compressible surface. We provide positive and negative examples, and we establish a simple sufficient condition. As an application, we obtain a new constructive proof of a result of Hatcher and Thurston about essential spanning surfaces for 2-bridge knots and links.
Comments32 pages, 34 figures. The first 19 pages of this paper were previously part of arXiv:2408.16948. Comments welcome!
