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Register a new userTaut sutured handlebodies as twisted homology products
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Authors
Margaret Nichols
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Friedl and Kim show any taut sutured manifold can be realized as a twisted homology product, but their proof gives no practical description of how complicated the realizing representation needs to be. We give a number of results illustrating the relationship between the topology of a taut sutured handlebody and the complexity of a representation realizing it as a homology product.
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We show that the sutured Khovanov homology of a balanced tangle in the product sutured manifold D x I has rank 1 if and only if the tangle is isotopic to a braid.
We show that the bordered-sutured Floer invariant of the complement of a tangle in an arbitrary 3-manifold $Y$, with minimal conditions on the bordered-sutured structure, satisfies an unoriented skein exact triangle. This generalizes a theorem by Manolescu for links in $S^3$. We give a theoretical proof of this result by adapting holomorphic polygon counts to the bordered-sutured setting, and also give a combinatorial description of all maps involved and explicitly compute them. We then show that, for $Y = S^3$, our exact triangle coincides with Manolescus. Finally, we provide a graded version of our result, explaining in detail the grading reduction process involved.
Morrison, Walker, and Wedrich used the blob complex to construct a generalization of Khovanov-Rozansky homology to links in the boundary of a 4-manifold. The degree zero part of their theory, called the skein lasagna module, admits an elementary definition in terms of certain diagrams in the 4-manifold. We give a description of the skein lasagna module for 4-manifolds without 1- and 3-handles, and present some explicit calculations for disk bundles over $S^2$.
Suppose $(M, gamma)$ is a balanced sutured manifold and $K$ is a rationally null-homologous knot in $M$. It is known that the rank of the sutured Floer homology of $Mbackslash N(K)$ is at least twice the rank of the sutured Floer homology of $M$. This paper studies the properties of $K$ when the equality is achieved for instanton homology. As an application, we show that if $Lsubset S^3$ is a fixed link and $K$ is a knot in the complement of $L$, then the instanton link Floer homology of $Lcup K$ achieves the minimum rank if and only if $K$ is the unknot in $S^3backslash L$.
We study 4-dimensional homology cobordisms without 3-handles, showing that they interact nicely with Thurston geometries, character varieties, and instanton and Heegaard Floer homologies. Using these, we derive obstructions to such cobordisms, with topological applications.
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