Knot homologies in monopole and instanton theories via sutures

In this paper we construct possible candidates for the minus versions of monopole and instanton knot Floer homologies. For a null-homologous knot $K \subset Y$ and a base point $p \in K$, we associate the minus versions, $\underline{\mathrm{KHM}}^- (Y,K,p)$ and $\underline{\mathrm{KHI}}^- (Y,K,p)$, to the triple $(Y,K,p)$. We prove that a Seifert surface of $K$ induces a $\mathbb{Z}$-grading, and there is an $U$-map on the minus versions, which is of degree $-1$. We also prove other basic properties of them. If $K \subset Y$ is not null-homologous but represents a torsion class, then we can also construct the corresponding minus versions for $(Y,K,p)$. We also proved a surgery-type formula relating the minus versions of a knot $K$ with those of the dual knot, when performing a Dehn surgery of large enough slope along $K$. The techniques developed in this paper can also be applied to compute the sutured monopole and instanton Floer homologies of any sutured solid tori.

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Received 16 April 2019

Accepted 25 March 2021

Published 8 June 2022