Upsilon-like concordance invariants from $\mathfrak{sl}_n$ knot cohomology

Abstract

We construct smooth concordance invariants of knots $K$ which take the form of piecewise linear maps ${\gimel }_n\left(K\right):\left[0,1\right]\to ℝ$ for $n\ge 2$. These invariants arise from ${\mathfrak{s}\mathfrak{l}}_n$ knot cohomology. We verify some properties which are analogous to those of the invariant $\Upsilon$ (which arises from knot Floer homology), and some which differ. We make some explicit computations and give some topological applications.

Further to this, we define a concordance invariant from equivariant ${\mathfrak{s}\mathfrak{l}}_n$ knot cohomology which subsumes many known concordance invariants arising from quantum knot cohomologies.