Journal of Knot Theory and Its Ramifications

We describe an algorithm that for every given braid $B$ explicitly constructs a function $f:{ℂ}^2\to ℂ$ such that $f$ is a polynomial in $u$, $v$ and $\overline{v}$ and the zero level set of $f$ on the unit three-sphere is the closure of $B$. The nature of this construction allows us to prove certain properties of the constructed polynomials. In particular, we provide bounds on the degree of $f$ in terms of braid data.

We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most literal sense, i.e. giving the set of quandle colorings the structure of a small category which is unchanged by Reidemeister moves. We derive some new enhancements of the counting invariant from this quiver structure and show that the enhancements are proper with explicit examples.

We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual trivalent braids. We provide two versions for the Markov-type theorem: one uses an algebraic approach similar to the case of classical braids and the other one is based on $L$-moves.

In this paper, we consider infinitesimal bending of the second-order of curves and knots. The total curvature of the knot during the second-order infinitesimal bending is discussed and expressions for the first and the second variation of the total curvature are given. Some examples aimed to illustrate infinitesimal bending of knots are shown using figures. Colors are used to illustrate curvature values at different points of bent knots and the total curvature is numerically calculated.

In this paper, we prove that given two cubical links of dimension two in ${ℝ}^4$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister and Roseman moves for classical tame knots (links) of dimensions one and two, respectively.

We show that if a braid $B$ can be parametrized in a certain way, then the previous work (B. Bode and M. R. Dennis, Constructing a polynomial whose nodal set is any prescribed knot or link, arXiv:1612.06328) can be extended to a construction of a polynomial $f:{ℝ}^4\to {ℝ}^2$ with the closure of $B$ as the link of an isolated singularity of $f$, showing that the closure of $B$ is real algebraic. In particular, we prove that closures of squares of strictly homogeneous braids and certain lemniscate links are real algebraic. We also show that the constructed polynomials satisfy the strong Milnor condition, providing an explicit fibration of the complement of the closure of $B$ over $S^1$.

Let $K$ be a hyperbolic knot in the $3$-sphere. If a $p/q$-Dehn surgery on $K$ produces manifold with an embedded Klein bottle or essential $2$-torus, then we prove that $\left|p\right|\le 8g_K-3$, where $g_K$ is the genus of $K$. We obtain different upper bounds according to the production of a Klein bottle, a non-separating $2$-torus, or an essential and separating $2$-torus. The well known examples which are the figure eight knot and the pretzel knot $K(-2,3,7)$ reach the given upper bounds. We study this problem considering null-homologous hyperbolic knots in compact, orientable and closed $3$-manifolds.

In this paper, we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan’s deformation. These invariants, called $\beta$-invariants, are essentially equivalent to Lipshitz, Ng, and Sarkar’s invariants ${\psi }^{\pm }$. From the $\beta$-invariants, we extract two non-negative integers which are transverse invariants (the $c$-invariants). Finally, we give several conditions which imply the non-effectiveness of the $c$-invariants, and use them to prove several vanishing criteria for the Plamenevskaya invariant $[\psi ]$, and the non-effectiveness of the vanishing of $[\psi ]$, for all prime knots with less than 12 crossings.

A relation between the dilatation of pseudo-Anosov braids and fixed point theory was studied by Ivanov. In this paper, we reveal a new relationship between the above two subjects by showing a formula for the dilatation of pseudo-Anosov braids by means of the representations of braid groups due to Jiang and Zheng.

We consider a problem of realizability of Gauss diagrams by closed plane curves where the plane curves have only double points of transversal self-intersection. We formulate the necessary and sufficient conditions for realizability. These conditions are based only on the parity of double and triple intersections of the chords in the Gauss diagram.

It was previously shown by the first author that every knot in $S^3$ is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot $K$ to be the minimum volume of any link $L$ in a natural class of alternating, hyperbolic links such that $K$ is ambient isotopic to a component of $L$. Our main result shows that the alternating volume of a knot is coarsely equivalent to the twist number of a knot.