#### Volume 23, issue 2 (2019)

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

### Lukas Lewark and Andrew Lobb

Geometry & Topology 23 (2019) 745–780
##### 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.

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##### Keywords
Khovanov–Rozansky cohomology, Knot concordance, Knot Floer homology
Primary: 57M25