arXiv subjects
Publications and source records attributed to J. González-Meneses.
We prove that the hypothetical extreme Khovanov cohomology of a link is the cohomology of the independence simplicial complex of its Lando graph. We also provide a family of knots having as many non-trivial extreme Khovanov cohomology modules as desired, that is, examples of $H$-thick knots which are as far of being $H$-thin as desired.
Let D be a link diagram with n crossings, s_A and s_B its extreme states and |s_AD| (resp. |s_BD|) the number of simple closed curves that appear when smoothing D according to s_A (resp. s_B). We give a general formula for the sum |s_AD|+|s_BD| for a k-almost alternating diagram D, for any k, characterising this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary. When D is dealternator connected, the triangulation is especially simple, yielding |s_AD|+|s_BD|=n+2-2k. This gives a simple geometric proof of the upper bound of the span of the Jones polynomial for dealternator connected diagrams, a result first obtained by Zhu in 1997. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al in 1992, is obtained as a corollary. As a new application, we prove that the Turaev genus is equal to the number k of dealternator crossings for any dealternator connected diagram.