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- (L), where (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- ₄ (L), where 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.