Knot Floer homology and Seifert surfaces

Let K be a knot in S^3 of genus g and let n>0. We show that if rk HFK(K,g) < 2^{n+1} (where HFK denotes knot Floer homology), in particular if K is an alternating knot such that the leading coefficient a_g of its Alexander polynomial satisfies |a_g| <2^{n+1}, then K has at most n pairwise disjoint non-isotopic genus g Seifert surfaces. For n=1 this implies that K has a unique minimal genus Seifert surface up to isotopy.Comment: 4 pages, n=0 case correcte

The coarse geometry of the Kakimizu complex

We show that the Kakimizu complex of minimal genus Seifert surfaces for a knot in the 3-sphere is quasi-isometric to a Euclidean integer lattice $\mathbb Z^n$ for some $n \geq 0$.Comment: 12 pages. Improvements to the exposition made in version

On sutured Floer homology and the equivalence of Seifert surfaces

We study the sutured Floer homology invariants of the sutured manifold obtained by cutting a knot complement along a Seifert surface, R. We show that these invariants are finer than the "top term" of the knot Floer homology, which they contain. In particular, we use sutured Floer homology to distinguish two non-isotopic minimal genus Seifert surfaces for the knot 8_3. A key ingredient for this technique is finding appropriate Heegaard diagrams for the sutured manifold associated to the complement of a Seifert surface.Comment: 32 pages, 17 figure

Combinatorial distance between HNN decompositions of a group

AbstractLet G be a group, and suppose that an epimorphism λ:G→Z and a section τ:Z→G are given. Put t=τ(1), and let D denote the set of HNN decompositions of G which have λ as natural epimorphism onto Z and t as stable letter in common. Also Dfg denotes the set of all α=A∗B∈D so that both A and B are finitely generated. In this paper we will associate to each α ∈ D a nonnegative integer valued function vα:G→Z≤0. Using these functions, we define a ‘distance’ on Dfg which reflects combinatorial differences of the decompositions

On links with locally infinite {K}akimizu complexes

We show that the Kakimizu complex of a knot may be locally infinite, answering a question of Przytycki--Schultens. We then prove that if a link $L$ only has connected Seifert surfaces and has a locally infinite Kakimizu complex then $L$ is a satellite of either a torus knot, a cable knot or a connected sum, with winding number 0.Comment: 9 pages, 5 figures; v2 minor has minor changes incorporating referee's comments. To appear in Algebraic & Geometric Topolog

Homogeneous links, Seifert surfaces, digraphs and the reduced Alexander polynomial

We give a geometric proof of the following result of Juhasz. \emph{Let $a_g$ be the leading coefficient of the Alexander polynomial of an alternating knot $K$. If $|a_g|<4$ then $K$ has a unique minimal genus Seifert surface.} In doing so, we are able to generalise the result, replacing `minimal genus' with `incompressible' and `alternating' with `homogeneous'. We also examine the implications of our proof for alternating links in general.Comment: 37 pages, 28 figures; v2 Main results generalised from alternating links to homogeneous links. Title change