On the tau invariants in instanton and monopole Floer theories

We unify two existing approaches to the tau invariants in instanton and monopole Floer theories, by identifying ${\tau }_G$, defined by the second author via the minus flavors ${\underline{KHI}}^{-}$ and ${\underline{KHM}}^{-}$ of the knot homologies, with ${\tau }_G^{♯}$, 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 ${\underline{KHI}}^{-}$ and ${\underline{KHM}}^{-}$ for twist knots.