A SLICE CROMWELL INEQUALITY FOR HOMOGENEOUS LINKS

Abstract 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.