Journal of Knot Theory and Its Ramifications

Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all odd dimensions.

We enumerate all the prime θ-curves with up to seven crossings to confirm a table made by Litherland. These θ-curves are mutually distinguished by the Yamada polynomial.

It is known that a maximal collection of essential annuli in H₂ could contain exactly 1, or 2, or at most 3 annuli, and a maximal collection of essential annuli in H_n with n ≥ 3 can contains at most 4n - 5 annuli, and the bound is best possible. In the present paper, we show that a maximal collection of essential annuli in H_n with n ≥ 3 contains at least two annuli, and for each m, 2 ≤ m ≤ 4n - 5, there exists a maximal collection of essential annuli in H_n which contains exactly m annuli.

Typical examples of textile structures are separated by finite type invariants of knots in non-trivial (in particular, non-orientable) manifolds. A new series of such invariants is described.

We show that there is a unique Markov trace on the tower of Temperley–Lieb type quotients of Hecke algebras of Coxeter type E_n (for all n ≥ 6). We explain in detail how this trace may be computed easily using tom Dieck's calculus of diagrams. As applications, we show how to use the trace to show that the diagram representation is faithful, and to compute leading coefficients of certain Kazhdan–Lusztig polynomials.

Every torus knot can be represented as a Fourier-(1,1,2) knot which is the simplest possible Fourier representation for such a knot. This answers a question of Kauffman and confirms the conjecture made by Boocher, Daigle, Hoste and Zheng.

In particular, the torus knot T_p,q can be parameterized as

We investigate the following existence problem for rational functions: for a given collection Π of partitions of a number n to define whether there exists a rational function f of degree n for which Π is the branch datum. An important particular case when the answer is known is the one when the collection Π contains a partition consisting of a single element (in this case, the corresponding rational function is equivalent to a polynomial). In this paper, we provide a solution in the case when Π contains a partition consisting of two elements.