Knot homologies in monopole and instanton theories via sutures

Abstract

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 can 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.

55 pages, 14 figures. Correct typos in the new version. Comments are welcomed!