Upsilon-like concordance invariants from 𝔰𝔩n knot cohomology

Abstract

We construct smooth concordance invariants of knots K which take the form of piecewise linear maps ‫ג‬n(K) : [0, 1] → R for n ≥ 2. These invariants arise from sln knot cohomology. We verify some properties which are analogous to those of the invariant Υ (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 sln knot cohomology which subsumes many known concordance invariants arising from quantum knot cohomologies.2010 Mathematics Subject Classification. 57M25.

Connected Floer homology of covering involutions

“…If K is concordant to an alternating (or more generally to a quasialternating) knot, then HFB − red-conn (K) = 0. Somewhat more surprisingly, the same vanishing holds for torus knots (a phenomenon reminiscent to the behaviour of the extension of the Upsilon-invariant of [25] to the Khovanov setting given by Lewark-Lobb in [17]): Theorem 1.5. For the torus knot T p,q we have that HFB − red-conn (T p,q ) = 0.…”

Braids with as many full twists as strands realize the braid index

“…We delay most of the proofs of the statements in this section to the end of the paper (see the Appendix) for the following reasons: first, these proofs are somewhat long but standard arguments using language from knot concordance theory and do not constitute the core of the † Recently, Grigsby-Wehrli-Licata [20] and Lewark-Lobb [32] defined Υ-type invariants using annular Khovanov cohomology and higher sl N -Khovanov-Rozansky cohomologies, respectively. However, neither of these invariants fit the framework of an Υ-type invariant as needed here.…”

“…Recently, Grigsby–Wehrli–Licata and Lewark–Lobb defined ‐type invariants using annular Khovanov cohomology and higher ‐Khovanov–Rozansky cohomologies, respectively. However, neither of these invariants fit the framework of an ‐type invariant as needed here.…”

Amphichiral knots with large 4-genus

“…A knot K in S 3 is called strongly negative amphichiral if there exists an orientation reversing involution ϕ : S 3 → S 3 such that ϕ(K) = K. Many concordance invariants vanish on such knots, including the classical Tristram-Levine signature function [Lev69,Tri69] and more modern invariants coming from Heegaard Floer and Khovanov homology like the τ -invariant [OS03], ν + -invariant [HW16], Υ-invariant [OSS17], s-invariant [Ras10], s n -invariants [Lob09,Wu09], s #invariant [KM13], and ‫-ג‬invariant [LL19]. Notably, this list contains almost all known lower bounds on the 4-genus, or minimal genus of a (smoothly or locally flatly) embedded orientable surface in B 4 with boundary the given knot.…”