Journal of Knot Theory and Its Ramifications

When two virtual knot diagrams are virtually isotopic, there is a sequence of Reidemeister moves and virtual moves relating them. I introduced a polynomial $q_K(t)$ of a virtual knot diagram $K$ and gave lower bounds for the number of Reidemeister moves in deformation of virtually isotopic knot diagrams by using $q_K(t)$. In this paper, I introduce bridge diagrams and polynomials of virtual knot diagrams based on parity of crossings, and show that the polynomials give lower bounds for the number of the third Reidemeister moves. I give an example which shows that the result is distinguished from that obtained from $q_K(t)$.

A filtration of Soergel bimodules by virtual crossing bimodules extends to Rouquier’s complexes associated with braid words. We show that these complexes are invariant up to filtered homotopy with respect to the second Reidemeister move, and the filtration of the triply graded link diagram homology, constructed by Khovanov through the application of the Hochschild homology, is invariant under Markov moves. We also prove that the homotopy equivalence of the complexes of braid words related by the third Reidemeister move violates filtration by at most two units.

An oriented monotone knot diagram is a knot diagram such that one meets each crossing as an over-crossing first as one travels the diagram with the orientation by starting at a point on the diagram. In this paper, unoriented knot projections which are monotone with an orientation and any over/under information are characterized. Also, monotone diagrams which are monotone with exactly one orientation and unique basepoint are characterized. As an application, a necessary condition for a knot projection with reductivity four is given.

Taking advantage of a numerical invariant, we visualize a distribution of rational homology 3-spheres on a plane via the Casson–Walker–Lescop (CWL) invariant and observe several aspects of the distribution. In particular, we study the characteristics of the distribution of lens spaces as a fundamental family of rational homology 3-spheres with a way to yield a family of estimation for the Dedekind sum. The CWL invariant captures the finiteness of lens space surgeries along knots. According to the finiteness, for example, the CWL invariant determines possible lens spaces as the results of integral surgeries along a knot $K$ with $a_1(K)=12$.

For each positive integer $n$ we will construct a family of infinitely many hyperbolic prime knots with alternation number 1, dealternating number equal to $n$, braid index equal to $n+3$ and Turaev genus equal to $n$.

We give new upper bounds on the stable commutator lengths of Dehn twists along separating curves in the mapping class group of a closed oriented surface. The estimates of these upper bounds are $O(1/g)$, where $g$ is the genus of the surface.

Every classical knot is band-pass equivalent to the unknot or the trefoil. The band-pass class of a knot is a concordance invariant. Every ribbon knot, for example, is band-pass equivalent to the unknot. Here we introduce the long virtual knot concordance group ${𝒱}{𝒞}$. It is shown that for every concordance class $[K]\in {𝒱}{𝒞}$, there is a $J\in [K]$ that is not band-pass equivalent to $K$ and an $L\in [K]$ that is not band-pass equivalent to either the long unknot or any long trefoil. This is accomplished by proving that $v_{2,1}+v_{2,2}(mod2)$ is a band-pass invariant but not a concordance invariant of long virtual knots, where $v_{2,1}$ and $v_{2,2}$ generate the degree two Polyak group for long virtual knots.

For the alternating knots or links, mutations do not change the arc index. In the case of nonalternating knots, some semi-alternating knots or links have this property. We mainly focus on the problem of mutation invariance of the arc index for nonalternating knots which are not semi-alternating. In this paper, we found families of infinitely many mutant pairs/triples of Montesinos knots with the same arc index.

For a finite graph $\mathrm{\Gamma }$, let $G(\mathrm{\Gamma })$ be the right-angled Artin group defined by the complement graph of $\mathrm{\Gamma }$. We show that, for any linear forest $\mathrm{\Lambda }$ and any finite graph $\mathrm{\Gamma }$, $G(\mathrm{\Lambda })$ can be embedded into $G(\mathrm{\Gamma })$ if and only if $\mathrm{\Lambda }$ can be realized as a full subgraph of $\mathrm{\Gamma }$. We also prove that if we drop the assumption that $\mathrm{\Lambda }$ is a linear forest, then the above assertion does not hold, namely, for any finite graph $\mathrm{\Lambda }$, which is not a linear forest, there exists a finite graph $\mathrm{\Gamma }$ such that $G(\mathrm{\Lambda })$ can be embedded into $G(\mathrm{\Gamma })$, though $\mathrm{\Lambda }$ cannot be embedded into $\mathrm{\Gamma }$ as a full subgraph.

For an effectively $n$-colorable knot $K$, the palette number ${\mathrm{\text{C}}}_n^{\ast }(K)$ is the minimum number of distinct colors for all effective $n$-colorings of $K$. It is known that ${\mathrm{\text{C}}}_n^{\ast }(K)\ge 2+\lfloor {log}_{2}n\rfloor$ for any effectively $n$-colorable knot $K$. In this paper, we show that for any odd $n\ge 3$ and effectively $n$-colorable torus knot $K$ it holds that ${\mathrm{\text{C}}}_n^{\ast }(K)=2+\lfloor {log}_{2}n\rfloor$ namely, any effectively $n$-colorable torus knot has an effectively $n$-colored diagram with $2+\lfloor {log}_{2}n\rfloor$ colors.