On Computing Kauffman Bracket Polynomial of Montesinos Links

Abstract

It is well known that Jones polynomial (hence, Kauffman bracket polynomial) of links is, in general, hard to compute. By now, Jones polynomials or Kauffman bracket polynomials of many link families have been computed, see [4,[7][8][9][10][11]. In recent years, the computer algebra (Maple) techniques were used to calculate link polynomials for various link families, see [7,[12][13][14]. In this paper, we try to design a maple program to calculate the explicit expression of the Kauffman bracket polynomial of Montesinos links. We first introduce a family of "ring of tangles" links, which includes Montesinos links as a special subfamily. Then, we provide a closed-form formula of Kauffman bracket polynomial for a "ring of tangles" link in terms of Kauffman bracket polynomials of the numerators and denominators of the tangles building the link. Finally, using this formula and known results on rational links, the Maple program is designed.

Tutte polynomials of fan-like graphs with applications in benzenoid systems

“…For convenience, in the following we sometimes abbreviate T (G; x, y) into T (G). The following lemma is contained as a special case in the general splitting-formula [16] of the Tutte polynomial or 2-splitting formula for the Tutte polynomial of signed graphs [14].…”

Density of Roots of the Yamada Polynomial of Spatial Graphs

“…(7) in the disk |a| < ε, and therefore, there exists a root A of the Yamada polynomial R[C n (Θ s (∞ k + ))] in the ε-neighborhood of z 0 if |G(0)| ≤ 1. Now we consider G(0) as a function (10) f (z) = (−1) k−1 z −k (z + z −1 ) + (z + 1 + z −1 )z −2k (z + 1 + z −1 )(−1) k−1 z −k (z + z −1 ) + (z + 1 + z −1 )z −2k .…”

Zeros of the Jones Polynomial are Dense in the Complex Plane

“…Lemma 2.5 [16] Let G be the union of two signed graphs G 1 and G 2 having only two common vertices u 1 and u 2 . Let H 1 and H 2 be signed graphs obtained from G 1 and G 2 , respectively, by identifying u 1 and u 2 .…”