Journal of Knot Theory and Its Ramifications

Energy minimizing smooth knot configurations have long been approximated by finding knotted polygons that minimize discretized versions of the given energy. However, for most knot energy functionals, the question remains open on whether the minimum polygonal energies are "close" to the minimum smooth energies. In this paper, we determine an explicit bound between the Minimum-Distance Energy of a polygon and the Möbius Energy of a piecewise-C² knot inscribed in the polygon. This bound is written in terms of the ropelength and the number of edges and can be used to determine an upper bound for the minimum Möbius Energy for different knot types.

It is well known that Jones polynomial (hence, Kauffman bracket polynomial) of links is, in general, hard to compute. By now, Jones polynomials or Kauffman bracket polynomials of many link families have been computed, see [4, 7–11]. In recent years, the computer algebra (Maple) techniques were used to calculate link polynomials for various link families, see [7, 12–14]. In this paper, we try to design a maple program to calculate the explicit expression of the Kauffman bracket polynomial of Montesinos links. We first introduce a family of "ring of tangles" links, which includes Montesinos links as a special subfamily. Then, we provide a closed-form formula of Kauffman bracket polynomial for a "ring of tangles" link in terms of Kauffman bracket polynomials of the numerators and denominators of the tangles building the link. Finally, using this formula and known results on rational links, the Maple program is designed.

We initiate a new way to study presentations of the inverse braid monoid . Our method makes use of the monoid of all order-preserving partial permutations on an n-element chain. As an application we derive the Easdown–Lavers (2004) presentation of , and we also obtain new presentations for the symmetric inverse monoid and the monoid of all order-preserving partial braids with at most n strings.

The topological analysis of enzymes, an active research topic, has allowed the application of the tangle model of Ernst and Sumners to deduce the action mechanism of several enzymes, modeled as 2-string tangles. By first deriving some results in the theory of 3-braids, in this paper we analyze knotted and linked products of site-specific recombination mediated by the Gin DNA invertase, an enzyme that involves 3-string tangles. Provided that the 3-tangles involved are 3-braids, we determine four families of solutions to its action, two families for each of the directly and inversely repeated site cases. For each case, one of the given solutions had not previously been reported in the related literature. These solutions were found using a computer algorithm, based on our theoretical results, which allows one to solve tangle equations under the assumption that the product of two or more rounds of recombinations is known.

We give a family of linear inequalities which strictly estimate relation among the coefficients of the Alexander polynomials of alternating knots of genus two. We also give such families for positive knots of genus two, and for homogeneous knots of genus two. As an application, we determine the alternating knots of genus two such that the leading coefficients of them are less than or equal to three.

In this paper, we present the Goeritz matrix for checkerboard colorable virtual links or, equivalently, checkerboard colorable links in thickened surfaces S_g × [0, 1], which is an extension of the Goeritz matrix for classical knots and links in ℝ³. Using this, we show that the signature, nullity and determinant of classical oriented knots and links extend to those of checkerboard colorable oriented virtual links.