The Turaev genus of an adequate knot (Q1032913)

The author shows in this paper that if a knot \(K\) admits an adequate diagram \(D\), then the Turaev genus of the knot \(K\) is the Turaev genus of the diagram \(D\), and they are equal to \(w_{Kh}(K)-2\), where \(w_{Kh}(K)\) is the width of the Khovanov homology of \(K\). Furthermore, if \(K'\) is a knot which has a mutant of \(D\), then the Turaev genus of \(K'\) is the same as that of \(K\). The connected sum of two adequate knots is adequate, and the author shows that the Turaev genus of the connected sum of two adequate knots is the sum of the Turaev genera of these two knots. The author asks whether the last two results could be true for all knots. Finally it is shown that an \(n\)-semi-alternating knot has Turaev genus \(n\), this gives the first examples of adequate knots of genus greater than 1.