A homology theory for framed links in I-bundles using embedded surfaces (Q2378303)

In this article the author defines a homology theory for framed links in I-bundles over an orientable surface \(F\), using the ideas of \textit{D. Bar-Natan} [Algebr. Geom. Topol. 9, 1443--1499 (2005; Zbl 1084.57011)]. Given a diagram \(D\) of the link in \(F\), the author defines the notion of state surface with respect to \(D\) as an embedded, orientable, compact surface in \(F\times I\) which has a state of \(D\) as its boundary in the top \((F\times\{0\})\) and essential oriented circles as its boundary in the bottom \((F\times\{1\})\) and satisfies some additional properties. The chain groups are defined by quotienting the free \(\mathbb Z\)-module generated by state surfaces by some fixed relations. After giving the definition of the boundary operator (and showing that its square equals zero) the author proves that the homology theory that he gets is equivalent to the link homology defined via decorated diagrams in \textit{M. M. Asaeda}, \textit{J. H. Przytycki} and \textit{A. S. Sikora} [Algebr. Geom. Topol. 4, 1177--1210 (2004; Zbl 1070.57008)]. As a corollary he obtains the invariance of his homology theory under Reidemeister moves 2 and 3, and proves that move 1 shifts the indices in a predictable way.