Genera of Knots in the Complex Projective Plane

Mr. Jake Pichelmeyer
Kansas State University
PhD Student (ABD)

“Let K be a knot and M be a smooth, closed four-dimensional manifold. The M-genus of K (plural: genera) is the least genus among all smooth, orientable surfaces S smoothly and properly embedded in punctured M such that the boundary of S is K. The M-genera have been computed for all 2,977 prime knots up to twelve crossings in the cases where M is the four-sphere or two-sphere cross two-sphere. In the case of the complex projective plane, there are 4,000+ prime knots up to twelve crossings along with their mirrors for which computation of the complex projective plane-genus is non-trivial. Of these, the complex projective plane-genus was known for only 8 such knots. We have obtained both obstruction and construction results that have allowed the computation of 146 more such prime knots of twelve crossings or less, along with several infinite families. We present background on this topic, explanation of how the constructions and obstructions were obtained, and how the computations were made using these results.”