The Non-Orientable 4-Genus of 11 Crossing Non-Alternating Knots

Abstract

The non-orientable 4-genus of a knot K in the three sphere is defined to be the minimum first Betti number of a non-orientable surface F in the four-ball so that K bounds F. We will survey the tools used to compute the non-orientable 4-genus, and use various techniques to calculate this invariant for non-alternating 11 crossing knots. We also will view obstructions to a knot bounding a M\"obius band given by the double branched cover of the three sphere branched over K.