Seifert surfaces distinguished by sutured Floer homology but not its Euler characteristic

Abstract

In this paper we find a family of knots with trivial Alexander polynomial, and construct two non-isotopic Seifert surfaces for each member in our family. In order to distinguish the surfaces we study the sutured Floer homology invariants of the sutured manifolds obtained by cutting the knot complements along the Seifert surfaces. Our examples provide the first use of sutured Floer homology, and not merely its Euler characteristic(a classical torsion), to distinguish Seifert surfaces. Our technique uses a version of Floer homology, called "longitude Floer homology" in a way that enables us to bypass the computations related to the SF H of the complement of a Seifert surface.

Bordered Floer homology and existence of incompressible tori in homology spheres

“…Since by Theorem 2.5, the map is an isomorphism. From here and by duality, is also an isomorphism and It follows from the proof of Proposition 5.3 from [Vaf15] (which may be extended to knots in arbitrary homology spheres) that This was also proved for fibered knots in [Eft05]. Thus, is trivial, is injective and is surjective.…”

Seifert surfaces in the 4-ball

“…There are many known examples of knots bounding genus-g incompressible Seifert surfaces that are not isotopic in S 3 (e.g. [Alf70, AS70, Dai73, Lyo74, Tro75, Eis77, Par78, Gab86, Kob89, Kak91, HJS13,Vaf15]). Most of these examples of distinct Seifert surfaces are known to become isotopic when their interiors are pushed into B 4 .…”

“…into B 4 (see [Rol76,p123]); the pretzel surfaces of [Tro75,Par78] (c.f. [Kob92]) were shown to be isotopic in B 4 by Livingston [Liv82, Section 6]; the surface families in [Eis77,Kak91] are obtained by twisting a surface many times about a satellite torus, which can be undone by isotopy in B 4 (see Lemma 3.5); the examples of [Gab86,Kob89,HJS13,Vaf15] arise from a Murasugi sum construction and are isotopic in B 4 via an isotopy involving pushing the plumbing region deeper into B 4 (see [Vaf15,Remark 3.1]). However, for one fairly well-known family of examples due to Lyon [Lyo74], there is no clear isotopy between the surfaces even when they are pushed into B 4 .…”

Boundary states of 4D topological matter: Emergence and full 3D-imaging of nodal Seifert surfaces

“…The linking properties of their homology generators can be used to compute [11,15] the Alexander polynomial -a classical knot invariant -of the NL or knot, hence distinguishing it from other nodal configurations. Indeed, Seifert surfaces are central to knot theory and low-dimensional topology [20], provoking many fascinating mathematical and computational problems, such as the uniqueness of a minimal genus Seifert surface [21], and their construction and visualization [22,23]. Its geometric appeal, e.g., appearing as a twisted band for the Hopf link, has also engendered much interest in other subfields, with alternative interpretations as contours of constant real space optical polarization azimuths [3] and dissipationless "Fermi" surfaces [24][25][26].…”