Journal of Knot Theory and Its Ramifications

The vertex distortion of a lattice knot is the supremum of the ratio of the distance between a pair of vertices along the knot and their distance in the ${\ell }_1$-norm. Inspired by Gromov, Pardon and Blair–Campisi–Taylor–Tomova, we show that results about the distortion of smooth knots hold for vertex distortion: the vertex distortion of a lattice knot is 1 only if it is the unknot, and there are minimal lattice-stick number knot conformations with arbitrarily high distortion.

We give a definition of an integer-valued function ${\sum }_i{\alpha }_i{x}_i^{\ast }$ derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free $\mathbb{Z}$-module generated by the arrow diagrams with at most $l$ arrows, called relators of Type ($\check{\mathrm{\text{I}}}$) (($\check{\mathrm{\text{SII}}}$), ($\check{\mathrm{\text{WII}}}$), ($\check{\mathrm{\text{SIII}}}$) or ($\check{\mathrm{\text{WIII}}}$), respectively), and introduce another function ${\sum }_i{\alpha }_i{\tilde{x}}_i^{\ast }$ to obtain ${\sum }_i{\alpha }_i{x}_i^{\ast }$. One of the main results shows that if ${\sum }_i{\alpha }_i{\tilde{x}}_i^{\ast }$ vanishes on finitely many relators of Type ($\check{\mathrm{\text{I}}}$) (($\check{\mathrm{\text{SII}}}$), ($\check{\mathrm{\text{WII}}}$), ($\check{\mathrm{\text{SIII}}}$) or ($\check{\mathrm{\text{WIII}}}$), respectively), then ${\sum }_i{\alpha }_i{x}_i^{\ast }$ is invariant under the deformation of type RI (strong RII, weak RII, strong RIII or weak RIII, respectively). The other main result is that we obtain new functions of arrow diagrams with up to six arrows explicitly. This computation is done with the aid of computers.

Tied links in $S^3$ were introduced by Aicardi and Juyumaya as standard links in $S^3$ equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces $L(p,1)$, in handlebodies of genus $g$, and in the complement of the $g$-component unlink. We introduce the tied braid monoids $T{M}_{g,n}$ by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we formulate and prove analogues of the Alexander and the Markov theorems for tied links in the 3-manifolds mentioned above. We also present an $L$-move braid equivalence for tied braids and we discuss further research related to tied links in knot complements and c.c.o. 3-manifolds. The theory of tied links has potential use in some aspects of molecular biology.

Let $L$ be a Montesinos link $M(-p,q,r)$ with positive rational numbers $p,q$ and $r$, each less than 1, and $c(L)$ the minimal crossing number of $L$. Herein, we construct arc presentations of $L$ with $c(L)$, $c(L)-1$ and $c(L)-2$ arcs under some conditions for $p$, $q$ and $r$. Furthermore, we determine the arc index of infinitely many Montesinos links.

We introduce a quandle invariant of classical and virtual links, denoted by $Q_{\mathrm{\text{tc}}}(L)$. This quandle has the property that $Q_{\mathrm{\text{tc}}}(L)\cong Q_{\mathrm{\text{tc}}}(L^{\prime })$ if and only if the components of $L$ and $L^{\prime }$ can be indexed in such a way that $L=K_1\cup \cdots \cup K_{\mu }$, $L^{\prime }=K_1^{\prime }\cup \cdots \cup K_{\mu }^{\prime }$ and for each index $i$, there is a multiplier ${𝜖}_i\in \{-1,1\}$ that connects virtual linking numbers over $K_i$ in $L$ to virtual linking numbers over $K_i^{\prime }$ in $L^{\prime }$: ${\ell }_{j/i}(K_i,K_j)={𝜖}_i{\ell }_{j/i}(K_i^{\prime },K_j^{\prime })$ for all $j\ne i$. We also extend to virtual links a classical theorem of Chen, which relates linking numbers to the nilpotent quotient $G(L)/G{(L)}_3$.

In this paper, we introduce an algebraic structure called a diquandle which is a set equipped with two quandle operations satisfying the right distributive laws. We discuss various examples and some properties of diquandles and also show that a diquandle enables us to distinguish oriented dichromatic links by telling that their coloring sets are different when their arcs are colored by elements of the diquandle.

According to the idea of Ozsváth, Stipsicz and Szabó, we define the knot invariant $\mathrm{{\rm Y}}$ without the holomorphic theory, using constructions from grid homology. We develop a homology theory using grid diagrams, and show that $\mathrm{{\rm Y}}$, as introduced this way, is a well-defined knot invariant. We reprove some important propositions using the new techniques, and show that $\mathrm{{\rm Y}}$ provides a lower bound on the unknotting number.

In this paper, we give two new criteria of detecting the checkerboard colorability of virtual links by using the odd writhe and the arrow polynomial of virtual links, respectively. As a result, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings.

In 2004, Neumann showed that the complex hyperbolic volume of a hyperbolic 3-manifold $M$ can be obtained as the image of the Dijkgraaf–Witten invariant of $M$ by a certain 3-cocycle. After that, Zickert gave an analogue of Neumann’s work for free fields containing finite fields. The author formulated a geometric method to calculate a weaker version of Zickert’s analogue, called the reduced Dijkgraaf–Witten invariant, for finite fields and gave a formula for twist knot complements and ${𝔽}_p$ in his previous work. In this paper, we show concretely how to calculate the reduced Dijkgraaf–Witten invariants of double twist knot complements and ${𝔽}_p$, and give a formula of them for $p=7$.

Cho and Murakami defined the potential function for a link $L$ in $S^3$ whose critical point, slightly different from the usual sense, corresponds to a boundary-parabolic representation $\rho :{\pi }_1(S^3\setminus L)\to {\mathrm{\text{PSL}}}_2(ℂ)$. They also showed that the volume and Chern–Simons invariant of $\rho$ can be computed from the potential function with its partial derivatives. In this paper, we extend the potential function to a representation that is not necessarily boundary-parabolic. We show that under a mild assumption it leads us to a combinatorial formula for computing the volume and Chern–Simons invariant of a ${\mathrm{\text{PSL}}}_2(ℂ)$-representation of a closed 3-manifold.