Volume 23, issue 2 (2019)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
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.

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