Abstract. We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.

We associate an open book with any connected plane checkerboard graph, thus providing a common extension of the classes of prime positive braid links and positive fibred arborescent links. As an application, we prove that the link type of a prime positive braid closure is determined by the linking graph associated with that braid.

It is well-known that generic perturbations of the complex Frobenius algebra used to define Khovanov cohomology each give rise to Rasmussen's concordance invariant s. This gives a concordance homomorphism to the integers and a strong lower bound on the smooth slice genus of a knot. Similar behavior has been observed in sl(n) Khovanov-Rozansky cohomology, where a perturbation gives rise to the concordance homomorphisms sn for each n ≥ 2, and where we have s 2 = s.We demonstrate that sn for n ≥ 3 does not in fact arise generically, and that varying the chosen perturbation gives rise both to new concordance homomorphisms as well as to new sliceness obstructions that are not equivalent to concordance homomorphisms.2010 Mathematics Subject Classification. 57M25.

We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebraic genus as a candidate for the optimal upper bound for the topological slice genus that is determined by the S-equivalence class of Seifert matrices.2010 Mathematics Subject Classification. 57M25, 57M27.

We construct smooth concordance invariants of knots K which take the form of piecewise linear maps ‫ג‬n(K) : [0, 1] → R for n ≥ 2. These invariants arise from sln knot cohomology. We verify some properties which are analogous to those of the invariant Υ (which arises from knot Floer homology), and some which differ. We make some explicit computations and give some topological applications.Further to this, we define a concordance invariant from equivariant sln knot cohomology which subsumes many known concordance invariants arising from quantum knot cohomologies.2010 Mathematics Subject Classification. 57M25.