On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology (Q1733106)

This paper shows a relation between the decategorification of Ozsváth-Szabó's theory to Viro's quantum relative \(\mathcal{A}^1\) of the Reshetikhin-Turaev function about the \(q\)-deformed universal enveloping superalgebra \(\mathcal{U}_a(\mathfrak{gl}(1\vert 1))\). Ozsváth-Szabó's theory can be summarized as follows. Suppose crossings, maximum points and minimum points of a knot projection are at different heights. One can use horizontal lines to separate the knot projection into elementary pieces, each of which contains exactly one of the crossings and max/min points. A horizontal line meets the knot projection at \(n\) oriented points. Ozsváth and Szabó define a differential graded algebra \(\mathcal{B}(n, \mathcal{S})\), where \(\mathcal{S}\) describes the orientation of points. To each elementary piece, they define a finitely generated \(DA\) bimodule over the algebras \(\mathcal{B}(n, \mathcal{S})\). Ozsváth and Szabó prove that the homology obtained by taking box tensor products of these bimodules over all elementary pieces coincides with knot Floer homology. Viro's quantum relative \(\mathcal{A}^1\) is defined for the same objects. For each elementary piece \(W\), there is a natural choice of \(1\)-palette \((B_W, M_W)\). The functor \(\mathcal{A}^1\) sends the bottom and top boundaries to two free \(B_W\)-modules, and sends \(W\) to a \(B_W\)-linear map between them. To consider the decategorification of Ozsváth-Szabó's theory, the author defines two subalgebras \(\mathcal{C}_r(n, \mathcal{S})\) and \(\mathcal{C}_l(n, \mathcal{S})\) of \(\mathcal{B}(n, \mathcal{S})\). He considers two types of triangulated categories of \(\mathcal{C}_r(n, \mathcal{S})\) and \(\mathcal{C}_l(n, \mathcal{S})\). The main result is as follows. Let \(Y\) be the set of \(n\) oriented points on a horizontal line. There are identifications of the Grothendieck groups (which are regarded as the decategorifications) of the triangulated categories mentioned above and the free module defined by \(\mathcal{A}^1\). The identifications are given by making a special choice of bases. For a tangle \(\Gamma\) that is not the terminal minimum, the decategorification of Ozsváth-Szabó's \(DA\) module agrees with the map defined by \(\mathcal{A}^1\) under the identifications above (a slight change of \(\Gamma\) is needed).