arXiv subjects

Stephan Martin Wehrli

Publications and source records attributed to Stephan Martin Wehrli.

3 recordsLinked to original sources

arXiv2023

On unification of colored annular sl(2) knot homology

We show that the Khovanov and Cooper-Krushkal models for colored sl(2) homology are equivalent in the case of the unknot, when formulated in the quantum annular Bar-Natan category. Again for the unknot, these two theories are shown to be equivalent to a third colored homology theory, defined using the action of Jones-Wenzl projectors on the quantum annular homology of cables. The proof is given by conceptualizing the properties of all three models into a Chebyshev system and by proving its uniqueness. In addition, we show that the classes of the Cooper-Hogancamp projectors in the quantum horizontal trace coincide with those of the Cooper-Krushkal projectors on the passing through strands. As an application, we compute the full quantum Hochschild homology of Khovanov's arc algebras. Finally, we state precise conjectures formalizing cabling operations and extending the above results to all knots.

math.GT↗

arXiv2019

On the functoriality of sl(2) tangle homology

We construct an explicit equivalence between the (bi)category of gl(2) webs and foams and the Bar-Natan (bi)category of Temperley-Lieb diagrams and cobordisms. With this equivalence we can fix functoriality of every link homology theory that factors through the Bar-Natan category. To achieve this, we define web versions of arc algebras and their quasi-hereditary covers, which provide strictly functorial tangle homologies. Furthermore, we construct explicit isomorphisms between these algebras and the original ones based on Temperley-Lieb cup diagrams. The immediate application is a strictly functorial version of the Beliakova-Putyra-Wehrli quantization of the annular link homology.

math.AT↗

arXiv2018

Quantum Link Homology via Trace Functor I

Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a~pair: bicategory $\mathbf{C}$ and endobifunctor $Σ\colon \mathbf C \to\mathbf C$. For a graded linear bicategory and a fixed invertible parameter $q$, we quantize this theory by using the endofunctor $Σ_q$ such that $Σ_q α:=q^{-°α}Σα$ for any 2-morphism $α$ and coincides with $Σ$ otherwise. Applying the quantized trace to the~bicategory of Chen-Khovanov bimodules we get a new triply graded link homology theory called quantum annular link homology. If $q=1$ we reproduce Asaeda-Przytycki-Sikora (APS) homology for links in a thickened annulus. We prove that our homology carries an action of $\mathcal U_q(\mathfrak{sl}_2)$, which intertwines the action of cobordisms. In particular, the~quantum annular homology of an $n$-cable admits an action of the braid group, which commutes with the quantum group action and factors through the Jones skein relation. This produces a nontrivial invariant for surfaces knotted in four dimensions. Moreover, a direct computation for torus links shows that the rank of quantum annular homology groups does depend on the quantum parameter $q$.

math.GT↗