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In Fall 2025, I supervised a group of undergraduate students in a project run through the Illinois Math Lab at UIUC. We investigated the geography and botany of nonorientable surfaces in B^4 with boundary torus knots. Our main goal was computational. We tested various examples with small non-orientable genus and normal Euler number pairs to determine if their Khovanov Jacobsson classes were non-zero. We eventually reached the following question: Are there classes in Khovanov homology of torus knots that are in the middle degrees that are represented geometrically (as the image of 1 under a map induced by a surface)? We began a project of writing code to randomly attach non-orientable bands to knot diagrams and compute the induced map using the recent software of Zsombor Feher.
The botany problem for non-orientable surfaces is a fairly uninvestigated topic and there are many simple questions and a dearth of examples. I hope to continus this research in future projects.Â
The botany problem for non-orientable surfaces is a fairly uninvestigated topic and there are many simple questions and a dearth of examples. I hope to continus this research in future projects.Â
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