Collaborative Research: RUI: Connecting Spatial Graphs to Links and 3-Manifolds

理解三维空间中打结物体的行为在数学和科学中具有根本的重要性，例如理解DNA。本计画探讨打结环与带有分支点的打结物件之间的关系。它将展示打结环的属性如何影响这些更复杂对象的属性，反之亦然。这些连接不仅将阐明打结物体的基本属性，还将阐明它们所处的三维空间。潜在的应用领域包括进一步理解三维和四维空间，以及DNA和打结聚合物。 本科生将为这个项目做出重大贡献，该项目在数学本科生和研究生之间建立了导师关系。该项目还支持一个夏令营，该夏令营利用艺术和数学游戏来培养小学生的基本计算能力，这些学生的数学成绩低于年级水平。PI开发了薄位置理论，以便阐明3-流形和结不变量的可加性或非可加性，如Heegaard亏格，桥数，隧道数和Gabai宽度。该项目将进一步开发这些工具，以便从节点和空间图的结构方面完全理解非可加性行为。这些技术也将被用来产生较低的界限上的某些概括的卫星结的桥数。此外，本发明还缝合流形理论将用于研究某些两点三边连接的打结空间图的补图的拓扑和几何结构。本项目由拓扑计划和刺激竞争研究既定计划（EPSCoR）联合资助。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评估被认为值得支持。影响审查标准。

Understanding the behavior of knotted objects in 3-dimensional spaces is of fundamental importance in mathematics and the sciences, for example in understanding DNA. This project investigates the relationship between knotted loops and knotted objects with branch points. It will show how the properties of knotted loops affect the properties of these more complicated objects and vice versa. These connections will elucidate fundamental properties not only of knotted objects but also the 3-dimensional spaces in which they reside. Potential areas of application include furthering understanding 3-dimensional and 4-dimensional spaces, as well as DNA and knotted polymers. Undergraduate students will make significant contributions to this project and the project builds mentorship connections between undergraduate and graduate students in mathematics. This project also supports a summer camp that uses the arts and math games to build basic numeracy skills in elementary school children who test below grade level in mathematics.The PIs have developed the theory of thin position so that it illuminates the additivity or non-additivity of 3-manifold and knot invariants such as Heegaard genus, bridge number, tunnel number, and Gabai width. This project will further develop these tools so that non-additivity behavior can be completely understood in terms of the structure of knots and spatial graphs. These techniques will also be used to produce lower bounds on the bridge number of certain generalizations of satellite knots. Additionally, sutured manifold theory techniques will be used to study the topological and geometric structure of the complements of certain knotted spatial graphs of two vertices joined by three edges.This project is jointly funded by Topology program and the Established Program to Stimulate Competitive Research (EPSCoR).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.