A family of transverse link homologies (Q258831)

An oriented tangent plane distribution \(\xi\) on an oriented 3-manifold \(M\) is called a contact structure on \(M\) if there is a contact 1-form \(\alpha\) satisfying \(\xi=\ker\alpha\), \(d\alpha_{|\alpha}>0\), and \(\alpha\wedge d\alpha>0\). The standard contact structure \(\xi_{\text{st}}\) on \(S^3\) is given by the contact form \(\xi_{\text{st}}=dz-ydx+xdy=dz+r^2d\theta\). An oriented smooth link \(L\) in \(S^3\) is called transverse if \({\alpha_{\text{st}}}_{|L}>0\). Two transverse links are said to be transverse isotopic if there is an isotopy from one to the other through transverse links. Two closed braids represent the same smooth link if and only if one of them can be changed into the other by a finite sequence of Markov moves: braid group relations, conjugations, and stabilizations and destabilizations. In [J. Knot Theory Ramifications 12, No. 7, 905--913 (2003; Zbl 1046.57007)], \textit{S. Yu. Orevkov} and \textit{V. V. Shevchishin} proved that two transverse closed braids are transverse isotopic if and only if the braid word of one of them can be changed into that of the other by a finite sequence of braid group relations, conjugations, and positive stabilizations and destabilizations.NEWLINENEWLINEIn this paper, the author defines a homology \(\mathcal{H}_N\) for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential \(ax^{N+1}\). In [Geom. Topol. 12, No. 3, 1387--1425 (2008; Zbl 1146.57018)], \textit{M. Khovanov} and \textit{L. Rozansky} studied the homology defined by the potential polynomial \(\sum\limits_{l=1}^{N+1}a_lx^l\in\mathbb Q[a_1,\dots,a_{N+1},x]\) which generalizes the HOMFLYPT homology. It turns out that \(\mathcal{H}_0\) is this HOMFLYPT homology up to a grading shift. The author proves that for \(N\geq 1\) if \(B\) is a closed braid and \((\mathcal{C}N(B), d_{mf}, d_{\chi})\) is the chain complex of matrix factorizations associated to \(B\), then the homotopy type \(\mathcal{H}_N\) of \(\mathcal{C}_N(B)\) does not change under transverse Markov moves. Also, it is shown that the homotopy equivalences induced by transverse Markov moves preserve the \(\mathbb{Z}_2\oplus\mathbb{Z}^{\oplus 3}\)-grading of \(\mathcal{C}_N(B)\), where the \(\mathbb Z_2\)-grading is the \(\mathbb Z_2\)-grading of the underlying matrix factorization and the three \(\mathbb Z\)-gradings are the homological, \(a\)- and \(x\)-gradings of \(\mathcal{C}_N(B)\). Consequently, for the homology \(\mathcal{H}_N(B)=H(H(\mathcal{C}_N(B),d_{mf}),d_{\chi})\) of \(\mathcal{C}_N(B)\), every transverse Markov move on \(B\) induces an isomorphism of \(\mathcal{H}_N(B)\) preserving the \(\mathbb{Z}_2\oplus\mathbb{Z}^{\oplus 3}\)-grading of \(\mathcal{H}_N(B)\) inherited from \(\mathcal{C}_N(B)\). He also discusses the decategorification of \(\mathcal{H}_N\) and the relation between \(\mathcal{H}_N\) and the \(\mathfrak{sl}(N)\) Khovanov-Rozansky homology.