Journal of Knot Theory and Its Ramifications

For a virtual knot $K$ and an integer $r\ge 0$, the $r$-covering $K^{(r)}$ is defined by using the indices of chords on a Gauss diagram of $K$. In this paper, we prove that for any finite set of virtual knots $J_0,J_2,J_3,\dots ,J_m$, there is a virtual knot $K$ such that $K^{(r)}=J_r(r=0\mathrm{\text{ and }}2\le r\le m)$, $K^{(1)}=K$, and otherwise $K^{(r)}=J_0$.

A formula for the Alexander polynomial of a 2-bridge knot or link given by Hartley and also by Minkus has a beautiful interpretation as a walk on the integers. We extend this to the 2-variable Alexander polynomial of a 2-component, 2-bridge link, obtaining a formula that corresponds to a walk on the 2-dimensional integer lattice.

We obtain information on torsion in Khovanov cohomology by performing calculations directly over $\mathbb{Z}/p^k\mathbb{Z}$ for $p$ prime and $k\ge 2$. In particular, we get that the torus knots $T(9,10)$ and $T(9,11)$ contain torsion of order $9$ and $27$ in their Khovanov cohomology.

A sequence of $F$-polynomials ${\{F_K^n(t,\ell )\}}_{n=1}^{\infty }$ of virtual knots $K$ was defined by Kaur et al. in 2018. These polynomials have been expressed in terms of index value of crossing and $n$-writhe of $K$. By the construction, $F$-polynomials are generalizations of Kauffman’s Affine Index Polynomial, and are invariants of virtual knot $K$. We present values of $F$-polynomials of oriented virtual knots having at most four classical crossings in a diagram.

We prove that a version of the Thurston–Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere $(M,\xi )$, whenever $\xi$ is tight. More specifically, we show that the self-linking number of a transverse link $T$ in $(M,\xi )$, such that the boundary of its tubular neighborhood consists of incompressible tori, is bounded by the Thurston norm $\text{∥}{T\text{∥}}_T$ of $T$. A similar inequality is given for Legendrian links by using the notions of positive and negative transverse push-off.

We apply this bound to compute the tau-invariant for every strongly quasi-positive link in $S^3$. This is done by proving that our inequality is sharp for this family of smooth links. Moreover, we use a stronger Bennequin inequality, for links in the tight 3-sphere, to generalize this result to quasi-positive links and determine their maximal self-linking number.

We show an infinite family of satellite knots that can be unknotted by a single band move, but such that there is no band unknotting the knots which is disjoint from the satellite torus.

Laplacian matrices of signed graphs in surfaces $S$ are used to define module and polynomial invariants of $\mathbb{Z}/2$-homologically trivial links in $S\times [0,1]$. Information about virtual genus is obtained.

We extend results of Positive Twist Knots and Thickenings [W. George, Positive twist knots and thickenings, J. Knot Theory Ramifications 22 (2013) 1350046] to show that positive twist knots ${{𝒦}}_m$, for $m\ge 3$, satisfy the Uniform Thickness Property.

We determine the minimal number of colors for nontrivial $\mathbb{Z}$-colorings on the standard minimal diagrams of $\mathbb{Z}$-colorable torus links. Also included is a complete classification of such $\mathbb{Z}$-colorings, which are shown by using rack colorings on link diagrams.

The warping degree of an oriented knot diagram is the minimal number of crossing changes which are required to obtain a monotone diagram from the diagram. The minimal warping degree of a knot is the minimal value of the warping degree for all oriented minimal diagrams of the knot. In this paper, all prime alternating knots with minimal warping degree two are determined.