Journal of Knot Theory and Its Ramifications

Habiro showed that two knots $K_1$ and $K_2$ are related by a finite sequence of clasp-pass moves, if and only if they have the same value for Vassiliev invariants of type $<3$. Tsukamoto showed that, if two knots differ by a clasp-pass move then the values of the Vassiliev invariant $V_K^{‴}(1)$ of degree $3$ for the two knots differ by $\pm 36$ or $0$, where $V_K(t)$ is the Jones polynomial of a knot $K$. If two virtual knots are related by clasp-pass moves, then they take the same value for all Vassiliev invariants of degree $<3$. We extend the Tsukamoto’s result to virtual knots by using a Vassiliev invariant $v_3$ of degree $3$, which is induced from the Kauffman polynomial. We also get a lower bound for the minimal number of clasp-pass moves needed to transform $K_1$ to $K_2$, if two virtual knots $K_1$ and $K_2$ can be related by a finite sequence of clasp-pass moves.

We prove a folklore theorem of Thurston, which provides necessary and sufficient conditions for primality of a certain class of theta-curves. Namely, a theta-curve in the 3-sphere with an unknotted constituent knot $\kappa$ is prime, if and only if lifting the third arc of the theta-curve to the double branched cover over $\kappa$ produces a prime knot. We apply this result to Kinoshita’s theta-curve.

We consider the problem of counting and of listing topologically inequivalent “planar” 4-valent maps with a single component and a given number $n$ of vertices. This enables us to count and to tabulate immersions of a circle in a sphere (spherical curves), extending results by Arnold and followers. Different options, where the circle and/or the sphere are/is oriented are considered in turn, following Arnold’s classification of the different types of symmetries. We also consider the case of bicolorable and bicolored maps or immersions, where faces are bicolored. Our method extends to immersions of a circle in a higher genus Riemann surface. There the bicolorability is no longer automatic and has to be assumed. We thus have two separate countings in nonzero genus, that of bicolorable maps and that of general maps. We use a classical method of encoding maps in terms of permutations, on which the constraints of “one-componentness” and of a given genus may be applied. Depending on the orientation issue and on the bicolorability assumption, permutations for a map with $n$ vertices live in $S_{4n}$ or in $S_{2n}$. In a nutshell, our method reduces to the counting (or listing) of orbits of certain subset of $S_{4n}$ (respectively, $S_{2n}$), under the action of the centralizer of a certain element of $S_{4n}$ (respectively, $S_{2n}$). This is achieved either by appealing to a formula by Frobenius or by a direct enumeration of these orbits. Applications to knot theory are briefly mentioned.

We prove that for particular infinite families of $L$-spaces, arising as branched double covers, the $d$-invariants defined by Ozsváth and Szabó assume arbitrarily large positive and negative values. As a consequence, we generalize a result by Greene and Watson by proving, for every odd number $\mathrm{\Delta }\ge 5$, the existence of infinitely many non-quasi-alternating homologically thin knots with determinant ${\mathrm{\Delta }}^2$, and a result by Hoffman and Walsh concerning the existence of hyperbolic weight $1$ manifolds, that are not surgery on a knot in $S^3$.

In this paper, we define and study flexible links and flexible isotopy in $ℝP^3\subset ℂP^3$. Flexible links are meant to capture the topological properties of real algebraic links. We classify all flexible links up to flexible isotopy using Ekholm’s interpretation of Viro’s encomplexed writhe.

We give a new interpretation of the Alexander polynomial ${\mathrm{\Delta }}_0$ for virtual knots due to Sawollek [On Alexander–Conway polynomials for virtual knots and Links, preprint (2001), arXiv:math/9912173] and Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000], and use it to show that, for any virtual knot, ${\mathrm{\Delta }}_0$ determines the writhe polynomial of Cheng and Gao [A polynomial invariant of virtual links, J. Knot Theory Ramifications 22(12) (2013), Article ID: 1341002, 33pp.] (equivalently, Kauffman’s affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22(4) (2013), Article ID: 1340007, 30pp.]). We also use it to define a second-order writhe polynomial, and give some applications.

We introduce a natural filtration in the space of knots and singular knots in the solid torus, and start the study of the type-two Vassiliev invariants with respect to this filtration. The main result of the work states that any such invariant within the second term of this filtration in the space of knots with zero winding number is a linear combination of seven explicitly described Gauss diagram invariants. This introduces a basis (and a universal invariant) for the type-two Vassiliev invariants for knots with zero winding number. Then we formalize the problem of exploring the set of all type-two invariants for knots with zero winding number.

We use virtual knot theory to detect the non-invertibility of some classical links in ${𝕊}^3$. These links appear in the study of virtual covers. Briefly, a virtual cover associates a virtual knot $\upsilon$ to a knot $K$ in a $3$-manifold $N$, under certain hypotheses on $K$ and $N$. Virtual covers of links in ${𝕊}^3$ come from taking $K$ to be in the complement $N$ of a fibered link $J$. If $J\bigsqcup K$ is invertible and $K$ is “close to” a fiber of $J$, then $\upsilon$ satisfies a symmetry condition to which some virtual knot polynomials are sensitive. We also discuss virtual covers of links $J\bigsqcup K$, where $J$ is not fibered, but is virtually fibered (in the sense of W. Thurston).