Link Floer homology and the Thurston norm

Abstract

We show that link Floer homology detects the Thurston norm of a link complement. As an application, we show that the Thurston polytope of an alternating link is dual to the Newton polytope of its multi-variable Alexander polynomial. To illustrate these techniques, we also compute the Thurston polytopes of several specific link complements.

The sutured Floer homology polytope

“…Moreover, y is nondegenerate if .M; / is reduced and horizontally prime. Using this norm, we can view Corollary 4.11 as an extension of a theorem of Ozsváth and Szabó [22] that link Floer homology detects the Thurston norm of the link complement. We will prove in [3] that if we symmetrize y , we get a seminorm that gives a lower bound on the seminorm defined by Scharlemann in [24], but is different from it.…”

Holomorphic disks, link invariants and the multi-variable Alexander polynomial

“…If one is not interested in this extra structure, but only b HFL as a graded group, then its construction is somewhat more elementary than the constructions described here. Properties of this invariant, and further computations, are given in [22].…”

The Thurston polytope for four-stranded pretzel links

“…This result offers no information about links with trivial Alexander polynomial, but work of Ozsváth-Szabó [6] and Ni [4] has shown that link Floer homology detects the Thurston norm. We prove Theorem 1 by determining the filtration support for the Heegaard Floer link homology of P q 1 ;r 1 ;q 2 ;r 2 .…”

“…The convex hull of these lattice points is known as the Newton polytope of L. A result of McMullen [3] relates B T .L/ to the Newton polytope, but in general .L/ does not determine B T .L/. Heegaard Floer link homology categorifies the multivariable Alexander polyonomial, and this stronger invariant determines the dual Thurston norm ball completely [6]. This relationship will be described precisely in Section 5.3.…”