Narrow Results

We use recently introduced Rasmussen invariant to find knots that are topologically locally-flatly slice but not smoothly slice. We note that this invariant can be used to give a combinatorial proof of the slice-Bennequin inequality. Finally, we compute the Rasmussen invariant for quasipositive knots and show that most of our examples of non-slice knots are not quasipositive and, to the best of our knowledge, were previously unknown.

The algebraic concordance group contains elements of order two, four, and of infinite order. Elements of infinite order are detected by the signature function. This paper develops computable invariants to simplify the computation of the order of torsion classes. The results are applied to determine the algebraic orders of all prime knots of 12 or fewer crossings.

Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this paper, we calculate the non-orientable 4-genus of knots with crossing number 10.

The non-orientable 4-genus of a knot $K$ in $S^3$ is defined to be the minimum first Betti number of a non-orientable surface $F$ smoothly embedded in $B^4$ 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 will also view obstructions to a knot bounding a Möbius band given by the double branched cover of $S^3$ branched over $K$.