Upsilon invariants from cyclic branched covers

We extend the construction of upsilon-type invariants to null-homologous knots in rational homology three-spheres. By considering $m$-fold cyclic branched covers with $m$ a prime power, this extension provides new knot concordance invariants $\Upsilon_m^C (K)$. We give computations of these invariants for some families of alternating knots and reprove some independence results.Comment: 26 pages, 13 figures. Comments are welcome

A cube of resolutions for knot Floer homology

We develop a skein exact sequence for knot Floer homology, involving singular knots. This leads to an explicit, algebraic description of knot Floer homology in terms of a braid projection of the knot.Comment: 55 pages, 24 figure