Given a diagram D of a knot K, we give easily computable bounds for Rasmussen's concordance invariant s(K). The bounds are not independent of the diagram D chosen, but we show that for diagrams satisfying a given condition the bounds are tight. As a corollary we improve on previously known Bennequin-type bounds on the slice genus.

We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2Z to a link diagram such that (1) the E 2 page of the resulting spectral sequence is naturally isomorphic to the Khovanov homology of the link; (2) this filtered complex behaves nicely under planar isotopy, disjoint union, and 1-handle addition; and (3) the spectral sequence collapses at the E 2 page for any diagram of the unlink. We prove that a Khovanov-Floer theory naturally yields a functor from the link cobordism category to the category of spectral sequences. In particular, every page (after E 1 ) of the spectral sequence accompanying a Khovanov-Floer theory is a link invariant, and an oriented cobordism in S 3 × [0, 1] between links in S 3 induces a map between each page of their spectral sequences, invariant up to smooth isotopy of the cobordism rel boundary.We then show that the spectral sequences relating Khovanov homology to Heegaard Floer homology and singular instanton knot homology are induced by Khovanov-Floer theories and are therefore functorial in the manner described above, as has been conjectured for some time. We further show that Szabó's geometric spectral sequence comes from a Khovanov-Floer theory, and is thus functorial as well. In addition, we illustrate how our framework can be used to give another proof that Lee's spectral sequence is functorial and that Rasmussen's invariant is a knot invariant. Finally, we use this machinery to define some potentially new knot invariants.

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.

There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the Lipshitz-Sarkar stable homotopy type and use it to make new computations.Similarly, there exists a simplified Khovanov-Rozansky sln complex for open 2-braids with oppositely oriented strands and an even number of crossings. Diagrams made by gluing tangles of this type are called matched diagrams, and knots admitting matched diagrams are called bipartite knots. To a pair consisting of a matched diagram and a choice of integer n ≥ 2, we associate a stable homotopy type. In the case n = 2 this agrees with the Lipshitz-Sarkar stable homotopy type of the underlying knot. In the case n ≥ 3 the cohomology of the stable homotopy type agrees with the sln Khovanov-Rozansky cohomology of the underlying knot.We make some consistency checks of this sln stable homotopy type and show that it exhibits interesting behaviour. For example we find a CP 2 in the sl 3 type for some diagram, and show that the sl 4 type can be interesting for a diagram for which the Lipshitz-Sarkar type is a wedge of Moore spaces. AL and DS were both supported by EPSRC grant EP/M000389/1, DJ was supported by an EPSRC graduate studentship.

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.