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On Floer minimal knots in sutured manifolds

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 Added by Zhenkun Li
 Publication date 2021
  fields
and research's language is English




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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 $S^3backslash L$.



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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.
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