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.

In this paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent of the corresponding slice-torus link invariant.