The Nonorientable Ribbon Conjecture, and Gordian Unknotting Number

不可定向丝带猜想和 Gordian 解结数

• 批准号: 1906413
• 负责人: Stanislav Jabuka
• 金额: $ 27.85万
• 依托单位: Board of Regents, NSHE, obo University of Nevada, Reno
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906413&HistoricalAwards=false
• 关键词: Nonorientable; Ribbon; Conjecture; Gordian; Unknotting

This project is jointly funded by the Topology program and the Established Program to Stimulate Competitive Research (EPSCoR).Knot theory - the study of how strings in 3-dimensional space can be knotted and unknotted - is a central tenet of low-dimensional topology, the study of spaces of dimension not larger than 4. Knots are used both to construct 3- and 4-dimensional spaces, and as a tool in their study. From its very inception over a century ago, surfaces have played an indispensable role in the study of knots, but only recently has this connection between the two broadened to include non-orientable surfaces. This project explores several novel connections between knots and the non-orientable surfaces they bound. While there has been a flurry of results in this direction in the past few years, still relatively little is known about their interplay, making this fertile ground for new research. A substantial portion of this proposal was inspired by problems stemming form DNA biology. Naturally occurring groups of enzymes in cells have the ability to either switch a crossing in a DNA strand, or to resolve that crossing. This naturally prompts the question of which knot types can become particularly simple after as few as possible alteration of these two types. While this research is not framed in terms of DNA biology, its results directly relate back to this question, and the PI expects his research to have application in the study of DNA recombination. Additionally, the research project will enhance visibility of mathematics in Nevada, and at the University of Nevada, Reno (UNR) in particular. This is especially relevant given the recently started Ph.D. program in mathematics at UNR, a program very much in its infancy. The award includes funding for graduate students.These research projects follow a wave of recent results involving the use of non-orientable surfaces in the study of knots. Some of this work had the potential of having been completed decades earlier, but a preference of topologists to work with orientable surfaces (a former preference of the PI as well) has left these stones unturned. This has dramatically changed in the last decade, and non-orientable surfaces are being recognized as powerful tools in low-dimensional topology.The research project concerns(i) Non-orientable Ribbon Conjecture, which compares embedded Moebius bands in the 4-ball with ribbon immersed M?bius bands in the 3-sphere.(ii) The Generalized Non-orientable Ribbon Conjecture which does the same for nonorientable surfaces of higher first Betti number.(iii) The notions of O-slicing number and O-concordance unknotting number, and consequences for the study of how knot concordance classes can change under O-moves on knot diagrams. (iv) Chiral smoothings of knots, a phenomenon by which a knot can be rendered equal to its own mirror image after a single crossing smoothing. The above problems fall into two broad categories. The first category aims to extend existing proofs of the (Orientable) Ribbon Conjecture for special families of knots, to the non-orientable setting, and touches on generalized analogues of both (Parts (i) and (ii)). The second category uses an entirely new approach to studying how local operations on knot diagrams impact the concordance class. The PI will study this problem for an infinite family of diagram moves, distinguished by the property that it includes many other local moves (Part (iii)).The tools used to study these problems contain aspects of gauge theory (of the Donaldson and Heegaard-Floer varieties), classical knot invariants, and explicit knot diagram modifications. Several of the research projects have aspects suitable for undergraduate and graduate student involvement.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目由拓扑学计划和刺激竞争研究的既定计划（EPSCoR）共同资助。纽结理论-研究三维空间中的弦如何打结和解开-是低维拓扑学的核心原则，研究维度不大于4的空间。结既可用于构建三维和四维空间，也可作为研究的工具。从一个世纪前开始，曲面就在纽结的研究中扮演着不可或缺的角色，但直到最近，这两者之间的联系才扩大到包括不可定向的曲面。 这个项目探讨了几个新的连接之间的结和不可定向的表面，他们绑定。虽然在过去的几年里，在这个方向上取得了一系列的成果，但对它们之间的相互作用仍然知之甚少，这为新的研究提供了肥沃的土壤。 这个提议的很大一部分是受到DNA生物学问题的启发。细胞中天然存在的酶组具有在DNA链中切换交叉或解决该交叉的能力。这自然引发了这样一个问题：在尽可能少地改变这两种类型之后，哪种类型的结可以变得特别简单。虽然这项研究不是在DNA生物学方面的框架，但其结果直接关系到这个问题，PI希望他的研究能够应用于DNA重组的研究。 此外，该研究项目将提高数学在内华达州，特别是在内华达州大学，里诺（UNR）的知名度。考虑到最近开始的博士学位，这一点尤其重要。UNR的一个数学项目，这个项目还处于起步阶段。该奖项包括对研究生的资助。这些研究项目遵循了最近一波涉及在结研究中使用不可定向表面的结果。其中一些工作有可能在几十年前就完成了，但拓扑学家倾向于使用可定向曲面（PI以前也倾向于使用），这使得这些石头没有转动。这已经发生了巨大的变化，在过去的十年中，和non-orientable表面正在被公认为在低维topology.The研究项目关注的强大工具（i）Non-orientable丝带猜想，比较嵌入Moebius带在4球与丝带浸入M？bius带在3-sphere中。(ii)广义不可定向带猜想，它对第一Betti数较高的不可定向曲面也有同样的作用。(iii)O-切片数和O-和谐解结数的概念，以及纽结图上的纽结和谐类在O-移动下如何变化的研究结果。 (iv)结的手征平滑，一种现象，通过这种现象，一个结可以被渲染为等于它自己的镜像后，一个单一的交叉平滑。 上述问题分为两大类。第一类的目的是扩展现有的证明（定向）丝带猜想的特殊家庭的结，非定向设置，并触及广义类似物（部分（i）和（ii））。第二类使用一种全新的方法来研究结图上的局部操作如何影响一致性类。PI将研究这个问题的无限族的图移动，区别在于它包括许多其他的本地移动的属性（第（iii）部分）。用于研究这些问题的工具包括规范理论（唐纳森和Heegaard-Floer品种），经典的结不变量，显式纽结图修改的方面。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。