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

Flora of Illinois, containing keys for the identification of the flowering plants and ferns, by George Neville Jones ...

no.2

Letter from William Croswell to John Jones, [1808]

Draft of a letter concerning Croswell's Mercator map.

Mesquakie (Fox) material culture : the William Jones and Frederick Starr collections / James W. VanStone.

n.s. no.30(1998)

Early Intermediate and Middle Horizon ceramic styles of the Cuzco Valley / Brian S. Bauer, Bradford M. Jones.

n.s. no.34(2003)

Notes/Observations on the Trinity by William Jones, ca. 1784

These appear to be Parmele's notes on a work by the scholar William Jones (1726-1800) regarding the Catholic doctrine of the Trinity.

Readings in mammalogy / selected from the original literature and introduced with comments by J. Knox Jones, Jr., and Sydney Anderson.

no.2 (1970)

Letters from T. Morton Jones to William Tudor,

Three letters and one price circular for patent wrought iron nails. Correspondence includes details on nail design and prices, as well as gossip about mutual friends and associates.