A geometric characterization of the upper bound for the span of the jones polynomial

Let D be a link diagram with n crossings, sA and sB be its extreme states and |sAD| (respectively, |sBD|) be the number of simple closed curves that appear when smoothing D according to sA (respectively, sB). We give a general formula for the sum |sAD| + |sBD| for a k-almost alternating diagram D, for any k, characterizing 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 |sAD| + |sBD| = 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 [On Kauffman brackets, J. Knot Theory Ramifications6(1) (1997) 125–148.]. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al. [Almost alternating links, Topology Appl.46(2) (1992) 151–165.], 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

The major allergens of birch pollen and cow milk, Bet v 1 and Bos d 5, are structurally related to human licocalin 2, enabling them to manipulate T-helper cells depending on their load with siderophore-bound iron

We conclude that Bet v 1 and Bos d 5 not only structurally mimic human LCN2, but also functionally by their ability to bind iron via siderophores. The apo-forms promote Th2 cells, whereas the holo-forms appear to be immunosuppressive. These results provide for the first time a functional understanding on the principle of allergenicity of major allergens from entirely independent sources, like birch and milk.