There are infinitely many pretzel links with the same Alexander polynomial (actually with trivial Alexander polynomial). By contrast, in this note we revisit the Jones polynomial of pretzel links and ...prove that, given a natural number S, there is only a finite number of pretzel links whose Jones polynomials have span S.
More concretely, we provide an algorithm useful for deciding whether or not a given knot is pretzel. As an application we identify all the pretzel knots up to nine crossings, proving in particular that 812 is the first non-pretzel knot.
In this paper, we derive formulae for the determinant of weaving knots W(3,n) and W(p,2). We calculate the dimension of the first homology group with coefficients in Z3 of the double cyclic cover of ...the 3-sphere S3 branched over W(3,n) and W(p,2) respectively. As a consequence, we obtain a lower bound of the unknotting number of W(3,n) for certain values of n.
Generalized Fishburn numbers and torus knots Bijaoui, Colin; Boden, Hans U.; Myers, Beckham ...
Journal of combinatorial theory. Series A,
February 2021, 2021-02-00, Volume:
178
Journal Article
Peer reviewed
Open access
Andrews and Sellers recently initiated the study of arithmetic properties of Fishburn numbers. In this paper we prove prime power congruences for generalized Fishburn numbers. These numbers are the ...coefficients in the 1−q expansion of the Kontsevich-Zagier series Ft(q) for the torus knots T(3,2t), t≥2. The proof uses a strong divisibility result of Ahlgren, Kim and Lovejoy and a new “strange identity” for Ft(q).
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