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Khovanov homology for virtual knots with arbitrary coefficients. (English. Russian original) Zbl 1142.57007

Izv. Math. 71, No. 5, 967-999 (2007); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 71, No. 5, 111-148 (2007).
One of the big problems in the theory of Khovanov homology [M. Khovanov, Duke Math. J. 101, No. 3, 359–426 (2000; Zbl 0960.57005)] is how to extend it from a theory which works on knots and links in \(\mathbb R^3\), to a full TFT, also categorifying quantum invariants of 3-manifolds, and many partial answers are known. Since the complex from which Khovanov homology is computed has its chains generated from a bifurcation cube of ”splittings” of a knot diagram, one step in this direction is to work with knots on higher genus surfaces, or even better with virtual knots. A virtual knot [L. H. Kauffman, Eur. J. Comb. 20, No.7,663–690 (1999; Zbl 0938.57006)] is an equivalence class of planar diagrams called virtual knot diagrams (4-valent graphs in which only some of the vertices need be endowed with additional over/under crossing information, the others remaining as virtual crossings) under a set of generalized Reidemeister moves. The paper under review produces a slightly different chain complex and differentials to generate Khovanov homology for virtual knots over arbitrary coefficient systems, in a way consistent with, but extending previous models. In the past there have been similar attempts, but because of problems with signs, the theories worked only over \(\mathbb Z/2\mathbb Z\).
Suppose \(K\) is a virtual knot diagram. By a state is meant a choice of smoothing at each classical crossing in \(K\). Each state determines a splitting of \(K\), which will be a collection of circles (say \(k\)), to which is associated a vector space which is the exterior product (not tensor/symmetric product) of \(k\) copies of a 2-dimensional vector space \(V\), along with a chosen basis formed from bases \(\{1,\pm{}X\}\) (signs dependent on orientations) for the \(k\) factors. This generates a complex of chains, \([[K]]\), doubly graded by height (number of \(B\)-smoothings in the state) and grading (height plus difference in numbers of 1’s and \(X\)’s in chain). States form a bifurcation cube, edges corresponding to changing the smoothing at one crossing between types \(A\) and \(B\). States joined by an edge have numbers of circles in their split diagrams, either identical or differing by one, as the affected circles may transform \(1\rightarrow1\), \(1\rightarrow2\) or \(2\rightarrow1\). A partial differential is defined corresponding to each classical crossing (the total differential being the sum of the partial differentials) by defining suitable maps \(\Delta:V\longrightarrow{}V\wedge{}V\), \(m:V\wedge{}V\longrightarrow{}V\) by \(\Delta(1)=1\wedge{}X_2+X_1\wedge1\), \(\Delta(X)=x_1\wedge{}X_2\), \(m(X_1\wedge{}X_2)=0\) while \(1\) behaves like a unit under the multiplication map \(m\). The main theorems are that the differential on the complex \({\mathcal C}(K)=[[K]]{n_+-2n_-}[-n_-]\) obtained from \([[K]]\) by shifting both the grading and height (where \(n_\pm\) denote the numbers of positively and negatively oriented crossings in \(K\)) defines a differential bigraded complex (that is \(\partial^2=0\)) whose homology groups are invariant under generalized Reidemeister moves and which, in the case that \(K\) is a virtual link diagram of an orientable atom, agree with the standard Khovanov homology. The methods of the paper can also be applied for ‘twisted virtual knots’ and the paper concludes with some remarks about generalizations of various results known for Khovanov homology of classical knots/links to the virtual cases.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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