Journal of Knot Theory and Its Ramifications

For a hyperbolic knot, an ideal triangulation of the knot complement corresponding to the colored Jones polynomial was introduced by Thurston. Considering this triangulation of a twist knot, we find a function which gives the hyperbolicity equations and the complex volume of the knot complement, using Zickert's theory of the extended Bloch group and the complex volume.

We also consider a formal approximation of the colored Jones polynomial. Following Ohnuki's theory of 2-bridge knots, we define another function which comes from the approximation. We show that this function is essentially the same as the previous function, and therefore it also gives the same hyperbolicity equations and the complex volume.

Finally we compare this result with our previous one which dealt with Yokota theory, and, as an application to Yokota theory, present a refined formula of the complex volumes for any twist knots.

We prove that every intrinsically knotted graph has at least 21 edges.

In 1983 Conway and Gordon and Sachs proved that every embedding of the complete graph on six vertices, K₆, is intrinsically linked. In 2004 it was shown that all straight-edge embeddings of K₆ have either one or three linked triangle pairs. We expand this work to characterize the straight-edge embeddings of K₇ and determine the number and types of links in every embedding which forms a convex polyhedron of seven vertices.

Relations will be described between the quandle cocycle invariant and the minimum number of colors used for non-trivial Fox colorings of knots and links. In particular, a lower bound for the minimum number is given in terms of the quandle cocycle invariant.

A 2-universal knot is a universal knot having all branching indices 1 or 2. Here we construct the first example of a 2-universal knot which could be a hyperbolic knot (SnapPea). Hence, we could have an example of a universal group generated by 180° rotations.

In this paper we resolve some conjectures concerning positive braid knots and almost alternating torus knots. Namely, we prove that the first Khovanov homology group of positive braid knot is trivial, as conjectured by Khovanov. Also, we generalize this result to show that the same is true in the case of Khovanov–Rozansky homology (sl(n) link homology) for any positive integer n. Moreover, by using the Khovanov homology theory, we prove the classical knot theory conjecture by Adams, that the only almost alternating torus knots are T_{3, 4} and T_{3, 5}.

It is a well known result that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module; we describe an alternating condition on elements of this skein module, show that it is satisfied by the Jones invariant of the single crossing tangles (⤲) and (⤲), and prove that it is preserved by appropriately "alternating" planar algebra compositions. Hence, this condition is satisfied by the Jones polynomial of all alternating tangles. Finally, in the case of 0-tangles, that is links, our condition is equivalent to simple alternation of the coefficients of the Jones polynomial.

By using a graph diagram named a magnetic graph diagram, we construct a polynomial invariant for knots and links. We show that it is a generalization of both the HOMFLY and the Kauffman polynomials.