Narrow Results

We calculate the highest and the lowest degrees of the Kauffman bracket polynomials of certain inadequate pretzel links and show that there is a knot K such that c(K) – r_deg V_K=k for any nonnegative integer k, where c(K) is the crossing number of K and r_deg V_K is the reduced degree of the Jones polynomial V_K of K.

We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Inoue’s paper [A. Inoue, Quasi-triviality of quandles for link-homotopy, J. Knot Theory Ramifications22(6) (2013) 1350026, doi:10.1142/S0218216513500260, MR3070837].

In this paper, we study conditions for $2$-component pretzel links to be concordant to the trivial link.

We give recursive relations to compute the rational Khovanov bigraded homology groups of three infinite families of non-quasi-alternating 3-column pretzel links. Based on this computations, we obtain an infinite family of homologically thin but not quasi-alternating knots.