On the tau invariants in instanton and monopole Floer theories

Abstract

We unify two existing approaches to the tau invariants in instanton and monopole Floer theories, by identifying ${\tau }_{\mathrm{G}}$, defined by the second author via the minus flavors ${\underline{\mathrm{KHI}}}^{-}$ and ${\underline{\mathrm{KHM}}}^{-}$ of the knot homologies, with ${\tau }_{\mathrm{G}}^{♯}$, defined by Baldwin and Sivek via cobordism maps of the $3$-manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute ${\underline{\mathrm{KHI}}}^{-}$ and ${\underline{\mathrm{KHM}}}^{-}$ for twist knots.

arXiv comment44 pages, 13 figures. Major revision with improved exposition and expanded introduction