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

了解三维空间中打结物体的行为对数学和科学具有重要意义，例如对DNA的理解。这个项目研究了结环和有分支点的结物之间的关系。它将展示结环的属性如何影响这些更复杂对象的属性，反之亦然。这些联系不仅阐明了打结物体的基本性质，而且阐明了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和结状聚合物。本科生将对该项目做出重大贡献，该项目在本科生和研究生之间建立了数学导师关系。该项目还支持一个夏令营，该夏令营使用艺术和数学游戏来培养数学测试低于年级水平的小学生的基本计算技能。pi发展了薄位置理论，从而阐明了3流形和结不变量（如Heegaard格、桥数、隧道数和Gabai宽度）的可加性或非可加性。本项目将进一步发展这些工具，使非可加性行为可以完全理解结点和空间图的结构。这些技术也将用于产生某些卫星节的泛化桥数的下界。此外，缝合流形理论技术将用于研究某些两个顶点由三条边连接的结空间图的补的拓扑和几何结构。该项目由拓扑学计划和促进竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.