相关论文: A formula for the Dubrovnik polynomial of rational…

We give an explicit formula for the HOMFLY polynomial of a rational link (in particular, a knot) in terms of a special continued fraction for the rational number that defines the given link.

We show that the problem of constructing a real rational knot of a reasonably low degree can be reduced to an algebraic problem involving the pure braid group: expressing an associated element of the pure braid group in terms of the…

We provide explicit formulas for the Alexander polynomial of pretzel knots and establish several immediate corollaries, including the characterization of pretzel knots with a trivial Alexander polynomial. As an application, we construct a…

We give a formula for Alexander polynomials of doubly primitive knots.

We find that Alexander polynomial of a ribbon knot in $ \mathbb{Z}HS^3 $ is determined by the intrinsic singularity information of its ribbon, and give a formula to calculate Alexander polynomial of a ribbon knot by that. We define half…

We describe a procedure that creates an explicit complex-valued polynomial function of three-dimensional space, whose nodal lines are the three-twist knot $5_2$. The construction generalizes a similar approach for lemniscate knots: a braid…

For arborescent links, we present an efficient method of computing their Alexander polynomials. Applying this method, we express the Alexander polynomials of Montesinos links in terms of certain functions associated to rational tangles…

We give a simple and practical algorithm to compute the link polynomials, which are defined according to the skein relations. Our method is based on a new total order on the set of all braid representatives. As by-product a new complete…

Twisted torus knots are a generalization of torus knots, obtained by introducing additional full twists to adjacent strands of the torus knots. In this article, we present an explicit formula for the Alexander polynomial of twisted torus…

We give an explicit formula for the Jones polynomial of any rational link in terms of the denominators of the canonical continued fraction of the slope of the given rational link.

We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial $\Delta_0$ (as defined by Silver and Williams) of these virtual twist knots. These results are applied…

We compute the Groebner basis of a system of polynomial equations related to the Jacobian conjecture using a recursive formula for the Catalan numbers.

We use Kauffman's bracket polynomial to define a complex-valued invariant of virtual rational tangles that generalizes the well-known fraction invariant for classical rational tangles. We provide a recursive formula for computing the…

In this note we give concise formulas, which lead to a simple and fast computer program that computes a powerful knot invariant. This invariant $\rho_1$ is not new, yet our formulas are by far the simplest and fastest: given a knot we write…

The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology of some sub-families of 3-strand pretzel knots, no general formula has been…

We give a simple proof of an explicit formula for Kerov polynomials. This formula is closely related to a formula of Goulden and Rattan.

The 2-loop polynomial is a polynomial presenting the 2-loop part of the Kontsevich invariant of knots. We show a cabling formula for the 2-loop polynomial of knots. In particular, we calculate the 2-loop polynomial for torus knots.

We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in \cite{Du} and a…

We give an algorithm for calculating the splitting type of the normal bundle of any rational monomial curve. The algorithm is obtained by reducing the calculus to a combinatorial problem and then by solving this problem.

We detail a procedure for the computation of the polynomial form of an electronic combinational circuit from the design equations in a truth table. The method uses the Buchberger algorithm rather than current traditional methods based on…