Knots, Disks, and Exotic Phenomena in Dimension 4

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Manifolds are objects which look locally simple but can have interesting global shape and properties. A guiding question in the field of topology is the classification, up to a given notion of equivalence, of all possible manifolds in a fixed dimension. Long-standing open questions concern the detection of exotic pairs in dimension 4. These are manifolds which are homeomorphic but not diffeomorphic, meaning that they are only distinguished by a very fine notion of equivalence. Techniques going back to Fox and Milnor and recently extended by Marengon, Manolescu and Piccirillo demonstrate that exotic pairs can be identified by using certain properties of knots, or entangled circles, which lie in the boundary of a 4-manifold after removing an open ball. The PI will explore this paradigm, which has the potential to lead to a fundamental shift in our ability to distinguish smooth 4-dimensional manifolds. An additional set of goals concerns studying the properties of knots in their own right, where certain old questions can be attacked by combinatorial tools introduced by the PI along with Blair and others. Overall, the development of these methods can have a lasting impact on the field of topology and on the applied sciences it informs, such as signal processing, data science and control theory, where manifolds appear in many guises - for one example, as state spaces of dynamical systems. The PI is also engaged in creating education and research opportunities for underrepresented communities. The PI will continue a program to employ combinatorial tools to address open questions in topology; her approach will provide graduate and advanced undergraduate students with an entry point to cutting-edge research in the field. The project outlines paths toward several distinct goals in low-dimensional topology for which the interaction between algebraic and geometric phenomena is the unifying principle; in which a combinatorial approach has proven effective; and in which knots in the 3-sphere and surfaces embedded in 4-manifolds are a central object of interest. Specific problems the PI will study include the Meridional Rank Conjecture of Cappell and Shaneson, the Slice-Ribbon Conjecture of Fox and Milnor and applications of knot theory to the detection of exotic smooth structures in dimension 4. The PI will employ state of the art tools, several of which she has previously helped develop and apply. The tools include: Coxeter quotients of knot groups; the Wirtinger number of a knot diagram; trisections of 4-manifolds; singular branched covers in dimension 4 and a ribbon obstruction extracted by the PI and collaborators from this context. 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.