Journal of Knot Theory and Its Ramifications

The Chen–Yang volume conjecture [Q. Chen and T. Yang, Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants, preprint (2015), arXiv:1503.02547] states that the growth rate of the Turaev–Viro invariants of a compact oriented $3$-manifold determines its simplicial volume. In this paper, we prove that the Chen–Yang conjecture is stable under $(2n+1,2)$-cabling.

In this paper, we prove that $w(K)=4w(J)$, where $w(.)$ is the width of a knot and $K$ is the Whitehead double of a nontrivial knot $J$.

In this paper, we define the rectangle condition for bridge decompositions of links in $S^3$ whose definition is analogous to the definition of the rectangle condition for Heegaard splittings of $3$-manifolds. We show that the satisfaction of the rectangle condition for an $n$-bridge decomposition can guarantee that the Hempel distance of the $n$-bridge decomposition is greater than or equal to two.

In this paper, we build a bridge between Conway–Coxeter friezes (CCFs) and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomial attached to rational links by using CCFs. As an application, one can give a complete invariant on CCFs of zigzag-type.

Let $\mathrm{\Gamma }$ be a chart, and we denote by ${\mathrm{\Gamma }}_m$ the union of all the edges of label $m$. A chart $\mathrm{\Gamma }$ is of type $(3,2,2)$ if there exists a label $m$ such that $w(\mathrm{\Gamma })=7$, $w({\mathrm{\Gamma }}_m\cap {\mathrm{\Gamma }}_{m+1})=3$, $w({\mathrm{\Gamma }}_{m+1}\cap {\mathrm{\Gamma }}_{m+2})=2$, and $w({\mathrm{\Gamma }}_{m+2}\cap {\mathrm{\Gamma }}_{m+3})=2$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(3,2,2)$.

For a knot or link ${𝒦}$, let $L({𝒦})$ be the ropelength of ${𝒦}$ and $\mathrm{\text{Cr}}({𝒦})$ be the crossing number of ${𝒦}$. In this paper, we show that there exists a constant $a>0$ such that $L({𝒦})\le a\mathrm{\text{Cr}}({𝒦}){ln}^5(\mathrm{\text{Cr}}({𝒦}))$ for any ${𝒦}$, i.e. the upper bound of the ropelength of any knot is almost linear in terms of its minimum crossing number. This result is a significant improvement over the best known upper bound established previously, which is of the form $L({𝒦})\le O(\mathrm{\text{Cr}}{({𝒦})}^{\frac{3}{2}})$. The proof is based on a divide-and-conquer approach on 4-regular plane graphs: a 4-regular plane graph of $n$ is first repeatedly subdivided into many small subgraphs and then reconstructed from these small subgraphs on the cubic lattice with its topology preserved with a total length of the order $O(n{ln}^5(n))$. The result then follows since a knot can be recovered from a graph that is topologically equivalent to a regular projection of it (which is a 4-regular plane graph).

The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number is algorithmically computable, it gives a more effective way to calculate an upper bound for the virtual bridge number from a virtual link diagram. As an application, we compute upper bounds for the virtual bridge numbers and the quandle counting invariants of virtual knots with 6 or fewer crossings. In particular, we found new examples of nontrivial virtual bridge number one knots, and by applying Satoh’s Tube map to these knots we can obtain nontrivial weakly superslice links.

We generalize the Wriggle polynomial, first introduced by L. Folwaczny and L. Kauffman, to the case of virtual tangles. This generalization naturally arises when considering the self-crossings of the tangle. We prove that the generalizations (and, by corollary, the original polynomial) are Vassiliev invariants of order one for virtual knots, and study some simple properties related to the connected sum of tangles.

For any virtual link $L=S\cup T$ that may be decomposed into a pair of oriented $n$-tangles $S$ and $T$, an oriented local move of type $T\mapsto \varphi (T)$ is a replacement of $T$ with the $n$-tangle $\varphi (T)$ in a way that preserves the orientation of $L$. After developing a general decomposition for the Jones polynomial of the virtual link $L=S\cup T$ in terms of various (modified) closures of $T$, we analyze the Jones polynomials of virtual links $L_1,L_2$ that differ via a local move of type $T\mapsto \varphi (T)$. Succinct divisibility conditions on $V(L_1)-V(L_2)$ are derived for broad classes of local moves that include the $\mathrm{\Delta }$-move and the double-$\mathrm{\Delta }$-move as special cases. As a consequence of our divisibility result for the double-$\mathrm{\Delta }$-move, we introduce a necessary condition for any pair of classical knots to be $S$-equivalent.

We classify Legendrian knots of topological type $7_6$ having maximal Thurston–Bennequin number confirming the corresponding conjectures of [W. Chongchitmate and L. Ng, An atlas of Legendrian knots, Exp. Math. 22(1) (2013) 26–37, arXiv:1010.3997].

If $L$ is a classical link then the multivariate Alexander quandle, $Q_A(L)$, is a substructure of the multivariate Alexander module, $M_A(L)$. In the first paper of this series, we showed that if two links $L$ and $L^{\prime }$ have $Q_A(L)\cong Q_A(L^{\prime })$, then after an appropriate re-indexing of the components of $L$ and $L^{\prime }$, there will be a module isomorphism $M_A(L)\cong M_A(L^{\prime })$ of a particular type, which we call a “Crowell equivalence.” In this paper, we show that $Q_A(L)$ (up to quandle isomorphism) is a strictly stronger link invariant than $M_A(L)$ (up to re-indexing and Crowell equivalence). This result follows from the fact that $Q_A(L)$ determines the $Q_A$ quandles of all the sublinks of $L$, up to quandle isomorphisms.