Mathematics Colloquia and Seminars

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The non-orientable 4-genus of a knot in the 3-sphere is defined as the smallest first Betti number of any non-orientable surface smoothly and properly embedded in the 4-ball, with boundary the given knot. This talk will summarize the short history of the non-orientable 4-genus, highlighting some recent results from Heegaard Floer homology. Then, a new approach to calculating this knot invariant will be described using Donaldson flavor gauge theory. We will show how to use this approach to compute the non-orientable 4-genus for all knots with crossing number 8 or 9, and we will contrast our approach with those from Heegaard Floer homology. Applications include proof of a conjecture of Murakami and Yasuhara. This is joint work with Tynan Kelly.