Cromwell [‘Homogeneous links’, J. London Math. Soc. (2) 39(3) (1989), 535–552] proved that the minimum v-degree of the HOMFLY polynomial of a homogeneous link L is bounded above by $1-\chi (L)$, where $\chi (L)$ is the maximum Euler characteristic of Seifert surfaces of L. We prove its slice version, stating that the minimum v-degree of the HOMFLY polynomial of a homogeneous link L is bounded above by $1-\chi _4(L)$, where $\chi _4(L)$ is the maximum four-dimensional Euler characteristic of L. As a byproduct, we prove a conjecture of Stoimenow [‘Some inequalities between knot invariants’, Internat. J. Math. 13(4) (2002), 373–393] that for an alternating link, the minimum v-degree of the HOMFLY polynomial is smaller than or equal to its signature.

We determine the locally flat cobordism distance between torus knots with small and large braid index, up to high precision. Here small means 2, 3, 4, or 6. As an application, we derive a surprising fact about torus knots that appear as cross-sections of almost minimal cobordisms between two-stranded torus knots and the trivial knot.

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 define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples where this invariant is a stronger obstruction to sliceness than the multivariable Levine–Tristram signature.