Extreme coefficients of Jones polynomials and graph theory

Abstract

We find families of prime knot diagrams with arbitrary extreme coefficients in their Jones polynomials. Some graph theory is presented in connection with this problem, generalizing ideas by Yongju Bae and Morton [3] and giving a positive answer to a question in their paper.

The Spread and Extreme Terms of Jones Polynomials

“…The coefficients of the maximal and minimal degree terms in the Jones polynomial may not in general be values of f , since the knot may not have a diagram for which they appear as the extreme coefficients. Since the original version of this paper was written there has been a nice construction due to Manchon [5] giving circle graphs (or equivalently the families of chords E and F ) which realise every integer value of f .…”

A geometric description of the extreme Khovanov cohomology

“…We want to remark that, although Theorem 9 holds for any simplicial complex, the graph obtained by the above procedure is not necessarily the Lando graph associated to a link diagram. A graph G is said to be realizable if there is a link diagram D such that G = G D (in [9] these graphs were originally called convertible).…”

A Geometric Characterization of the Upper Bound for the Span of the Jones Polynomial

“…m) above is the maximal (resp. minimal) possible degree of the Kauffman bracket D of an arbitrary diagram D. Moreover, the degree of any term in the Kauffman bracket is congruent with m (and also with M ) modulo 4 (see, for instance, [10]). In other words, the Kauffman bracket of any diagram D can be written as…”

“…[14, Theorem 4] If D is a dealternator connected, k-almost alternating diagram with n crossings, thenspan( D ) ≤ 4(n − k).Proof. It is well known[10] that if we denote M = n + 2|s A D| − 2 and m = −n−2|s B D|+2, then the maximal (resp. minimal) degree of the Kauffman bracket of the diagram D is at most M (resp.…”