Bar-Natan's geometric complex and a categorification of the Dye-Kauffman-Miyazawa polynomial (Q2788550)

Links diagrams on compact surfaces and their invariants under various conditions have been objects of study as a direction of further development of quantum invariant theory of links in \(S^3\). Virtual link diagrams are link diagrams ``drawn on paper'' with encircled crossings, introduced in [\textit{L. H. Kauffman}, Eur. J. Comb. 20, No. 7, 663--690, Art. No. eujc.1999.0314 (1999; Zbl 0938.57006)]. It has been proved in [\textit{J. S. Carter} et al., J. Knot Theory Ramifications 11, No. 3, 311--322 (2002; Zbl 1004.57007)] that there is a bijection between the equivalence classes of virtual link diagrams with respect to the generalized Reidemeister moves and the stable equivalence classes of link diagrams on compact surfaces. This means that invariants of virtual link diagrams give rise to those of link diagrams on compact surfaces. A generalization of Jones polynomials has been constructed by Dye, Kauffman and Miyazawa (DKMi-polynomials).NEWLINENEWLINEKhovanov constructed a categorification of the Jones polynomial, namely a homology theory which is a link invariant whose graded Euler characteristic coincides with the Jones polynomial. Categorifications of DKMi-polynomials have been constructed by introducing new gradings of various sorts, along the line of \textit{D. Bar-Natan} [Geom. Topol. 9, 1443--1499 (2005; Zbl 1084.57011)]. There the chain spaces are generated by algebras associated with diagrams of circles resulting from smoothing of crossing in link diagrams, and differentials are given by change of smoothings.NEWLINENEWLINEOn the other hand, Turaev introduced a notion of homotopy quantum field theories (HQFT) in [\textit{V .Turaev}, ``Homotopy field theory in dimension 2 and group-algebras'', Preprint, \url{arXiv:math/9910010}]. It is a Topological quantum field theory with additional structures. In this paper, the author gives a categorification of DKMi-polynomials using the unoriented HQFT, namely a triple graded homology theory which is an invariant of stable equivalence classes, whose graded Euler characteristics coincide with DKMi-polynomials.