Examples on polynomial invariants of knots and links
After the discovery of the Jones polynomial [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New. Ser. 12, 103-111 (1985; Zbl 0564.57006)], other new polynomial invariants of knots and links have been discovered; the 2- variable Jones polynomial of \textit{P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, J. Millett, A. Ocneanu} [ibid. 12, 239-246 (1985; Zbl 0572.57002)] and \textit{J. H. Przytycki} and \textit{P. Traczyk} [to appear in Kobe J. Math.], which generalizes both the Jones polynomial and the reduced Alexander polynomial, and is a skein invariant, and the Q or absolute polynomial of R. D. Brandt, W. B. R. Lickorish, K. C. Millett, and C. F. Ho. We first give infinitely many skein equivalent knots which are classified by the Q polynomial. Second, we give an example of arbitrarily many prime knots with the same Jones but distinct Alexander polynomials, where the \(''V_{\infty}\) formula'' for the Jones polynomial of Jones and Birman is used. Lastly, we consider 2-bridge knots and links: We construct arbitrarily many 2-bridge knots with the same Jones and Q polynomials, and arbitrarily many skein equivalent fibered 2-bridge links with the same 2-variable Alexander and Q polynomials. Also we give a pair of skein equivalent 2-bridge links with the same Q polynomial, but distinct 2- variable Alexander polynomials. And we give an example of a fibered 2- bridge link which is transformed into a non-isotopic link by reversing the orientation of one component, but whose skein equivalent class and 2- variable Alexander polynomial do not change.
scientific article; zbMATH DE number 3935986
scientific article