Heegaard Floer homology of some Mazur type manifolds

Abstract

Abstract. We show that an infinite family of contractible 4-manifolds have the same boundary as a special type of plumbing. Consequently their Ozsváth-Szabó invariants can be calculated algorithmically. We run this algorithm for the first few members of the family and list the resulting Heegaard-Floer homologies. We also show that the rank of the Heegaard-Floer homology can get arbitrary large values in this family by using its relation with the Casson invariant. For comparison, we list the ranks of Floer homologies of all the examples of Briekorn spheres that are known to bound contractible manifolds.

Corks, exotic 4-manifolds and knot concordance

“…A simple isotopy gives the third diagram, and the aforementioned handle moves give the last diagram of X Ln . Corollary 1.7 in [7] and the last diagram thus tells that the Casson invariant λ(∂X Ln ) of the boundary 3-manifold is equal to −2. Therefore each X Ln is not homeomorphic to D 4 .…”

Classical and new plumbed homology spheres bounding contractible manifolds

“…4 However, the above type of conjecture fails for non Seifert fibered plumbed homology spheres. The first examples were provided by Maruyama [Mar82], independently refound by Akbulut and Karakurt [AK14]. These manifolds are not Seifert fibered 5 unless n = 1 and they can be realized as the boundary of a plumbing graph with two nodes and five branches, and are reproven via our approach.…”

Heegaard Floer homology and several families of Brieskorn spheres

“…Note that the star-shaped plumbing graph associated to a Seifert manifold can always be drawn such that no vertices are bad except possibly the central one. We'll briefly describe the algorithm; see [AK12] for a very concrete user's guide. Let Γ be the plumbing graph and let X(Γ) denote the associated plumbed 4-manifold.…”

“…I would like to thank Ç agri Karakurt for introducing me to Némethi's graded roots during his visit to Rice in Spring 2012 (and he and his coauthor for the useful primer in [AK12]). I would also like to thank Tye Lidman for some helpful conversations.…”

“…Némethi provided a nice combinatorial framework for this description in [Ném05] via a very concrete algorithm: a plumbing graph leads to a function τ : Z ≥0 → Z, which in turn induces an intermediate object known as a "graded root"; this gadget both determines HF + and carries some extra data relevant to singularity theory. For a very approachable "user's guide" to Némethi's algorithm, see [AK12].…”