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基本的重要性，例如在理解DNA方面。这个项目研究打结环和带有分支点的打结对象之间的关系。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。这些联系不仅将阐明打结物体的基本性质，而且还将阐明它们所在的3维空间。潜在的应用领域包括进一步了解三维和四维空间，以及DNA和打结聚合物。本科生将对这个项目做出重大贡献，该项目在本科生和研究生之间建立了数学方面的导师关系。这个项目还支持一个夏令营，它使用艺术和数学游戏来培养那些数学成绩低于年级水平的小学生的基本算术技能。PI发展了薄位置理论，从而解释了3-流形和纽结不变量的可加性或非可加性，如Heegaard亏格、桥号、隧道号和Gabai宽度。这个项目将进一步开发这些工具，以便可以根据节点和空间图的结构完全理解非可加性行为。这些技术还将被用来产生某些卫星结的推广的桥数的下界。此外，缝合流形理论技术将被用来研究由三条边连接的两个顶点的某些纽结空间图的补图的拓扑和几何结构。该项目由拓扑学计划和既定的刺激竞争研究计划(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优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.