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

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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.