The Spread and Extreme Terms of Jones Polynomials

Abstract

We adapt Thistlethwaite's alternating tangle decomposition of a knot diagram to identify the potential extreme terms in its bracket polynomial, and give a simple combinatorial calculation for their coefficients, based on the intersection graph of certain chord diagrams.

Extreme coefficients of Jones polynomials and graph theory

“…This non-negative integers does not depend on the chosen order of the b(s) crossings, although the association of +1 and −1 to each one of the b(s) crossings does. Now we recall the connection between a M and graph theory given in [3]. In order to obtain the Lando's graph of D start with s A D and delete the A-chords joining two different components.…”

“…The example in Figure 4 is exhibited in [3], where the corresponding Lando's graph G D has f (G D ) = 3. The question formulated in [3] is if any integer n …”

“…The first approaching follows Yonju Bae and Morton [3]. In their paper a connection between a M and graph theory is given.…”

“…|C| where C runs over all the independent subsets of vertices of G, where independent means that there is no edge joining two vertices of C. In [3] it arises the question of if any integer can be realized as f (G) for some graph G D . We have found a positive answer to this question.…”

“…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. …”