Journal of Knot Theory and Its Ramifications

We introduce a map, which we call the bind map, from the set of all classical braids to that of all trivalent braids. Then we define a notation for handlebody-links with a pair of a bind map and a classical braid. We show that isotopies and braid relations are equivalent for trivial handlebody-braids obtained from classical 3-braids with bind maps. We introduce two types of graphs for a classical braid, which indicate how the binding forms of the trivalent braids are transformed each other. We determine all patterns of the graphs for classical 3-braids.

This paper gives a connection between well-chosen reductions of the Links–Gould invariants of oriented links and powers of the Alexander–Conway polynomial. This connection is obtained by showing the representations of the braid groups we derive the specialized Links–Gould polynomials from can be seen as exterior powers of a direct sum of Burau representations.

We consider some families of links which include $(-2,2m+1,2n)$-pretzel links and twisted Whitehead links. We calculate the ${\mathrm{\text{SL}}}_2(ℂ)$-character varieties of these families, and determine the number of irreducible components of these character varieties.

Using a special form of spanning surface for a knot, we give a formula for the coefficient of the $z^2$-term of the Alexander–Conway polynomial in terms of the sum of determinants of the blocks of $2\times 2$ submatrices of the Seifert matrix, from which the topological meaning of the coefficient is revealed.

It is well known that one of O’Hara’s knot energies is called the Möbius energy because of its invariance under Möbius transformations. We showed in a previous paper that the Möbius energy can be decomposed into three parts that retain invariance but we left open the question of invariance regarding inversions with respect to spheres centered on a knot. Here, we answer this question under the assumption that the knots have extra regularity. The result holds not only for knots but also for closed curves in ${ℝ}^n$.

We give two new criteria of detecting checkerboard colorability of virtual links in terms of the Sawollek polynomials and compare the criteria with known criteria. By using these criteria we determine checkerboard colorability of virtual knots up to four classical crossings, with six exceptions.

In this paper, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot $K$ with Alexander polynomial ${\mathrm{\Delta }}_K(t)$, I then use these lassos as patterns to construct families of satellite knots that have Alexander polynomial ${\mathrm{\Delta }}_K(t^d)$ where $d\in \mathbb{N}\cup \{0\}$. In particular, I prove that if $d\in \{0,1,2,3\}$ these satellite knots have different Jones polynomials.

Conjecture $\mathbb{Z}$ is a knot theoretical equivalent form of the Kervaire conjecture. We show that Conjecture $\mathbb{Z}$ is true for all the pretzel knots of the form $P(p,q,-r)$ where $p$, $q$ and $r$ are odd positive integers.

Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from a projection of the cobordism into 3-space by taking 2-dimensional cross-sections perpendicular to a fixed direction. We present a set of movie moves that are sufficient to connect any two movies of isotopic singular link cobordisms.