The Nonorientable Ribbon Conjecture, and Gordian Unknotting Number

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This project is jointly funded by the Topology program and the Established Program to Stimulate Competitive Research (EPSCoR).Knot theory - the study of how strings in 3-dimensional space can be knotted and unknotted - is a central tenet of low-dimensional topology, the study of spaces of dimension not larger than 4. Knots are used both to construct 3- and 4-dimensional spaces, and as a tool in their study. From its very inception over a century ago, surfaces have played an indispensable role in the study of knots, but only recently has this connection between the two broadened to include non-orientable surfaces. This project explores several novel connections between knots and the non-orientable surfaces they bound. While there has been a flurry of results in this direction in the past few years, still relatively little is known about their interplay, making this fertile ground for new research. A substantial portion of this proposal was inspired by problems stemming form DNA biology. Naturally occurring groups of enzymes in cells have the ability to either switch a crossing in a DNA strand, or to resolve that crossing. This naturally prompts the question of which knot types can become particularly simple after as few as possible alteration of these two types. While this research is not framed in terms of DNA biology, its results directly relate back to this question, and the PI expects his research to have application in the study of DNA recombination. Additionally, the research project will enhance visibility of mathematics in Nevada, and at the University of Nevada, Reno (UNR) in particular. This is especially relevant given the recently started Ph.D. program in mathematics at UNR, a program very much in its infancy. The award includes funding for graduate students.These research projects follow a wave of recent results involving the use of non-orientable surfaces in the study of knots. Some of this work had the potential of having been completed decades earlier, but a preference of topologists to work with orientable surfaces (a former preference of the PI as well) has left these stones unturned. This has dramatically changed in the last decade, and non-orientable surfaces are being recognized as powerful tools in low-dimensional topology.The research project concerns(i) Non-orientable Ribbon Conjecture, which compares embedded Moebius bands in the 4-ball with ribbon immersed M?bius bands in the 3-sphere.(ii) The Generalized Non-orientable Ribbon Conjecture which does the same for nonorientable surfaces of higher first Betti number.(iii) The notions of O-slicing number and O-concordance unknotting number, and consequences for the study of how knot concordance classes can change under O-moves on knot diagrams. (iv) Chiral smoothings of knots, a phenomenon by which a knot can be rendered equal to its own mirror image after a single crossing smoothing. The above problems fall into two broad categories. The first category aims to extend existing proofs of the (Orientable) Ribbon Conjecture for special families of knots, to the non-orientable setting, and touches on generalized analogues of both (Parts (i) and (ii)). The second category uses an entirely new approach to studying how local operations on knot diagrams impact the concordance class. The PI will study this problem for an infinite family of diagram moves, distinguished by the property that it includes many other local moves (Part (iii)).The tools used to study these problems contain aspects of gauge theory (of the Donaldson and Heegaard-Floer varieties), classical knot invariants, and explicit knot diagram modifications. Several of the research projects have aspects suitable for undergraduate and graduate student involvement.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.