Classical Knot Research Articles

Derivatives of Jones polynomials detect delta moves in virtual knots

For classical knots, [Formula: see text]-moves and crossing changes can both unknot any knot. But many virtual knots cannot be unknotted with these moves. Moreover, there are many virtual knots that can be unknotted by crossing changes but cannot be unknotted by [Formula: see text]-moves. We show that the derivatives of the Jones polynomial can detect whether a virtual knot can be unknotted by [Formula: see text]-moves.

Multivariate Alexander quandles, IV. The medial quandle of a link

Joyce observed that the Alexander invariant and the medial quandle of a classical knot are equivalent to each other, as invariants. In this paper, we discuss the rather complicated extension of Joyce’s observation to several different medial quandles and reduced (one-variable) Alexander modules associated with classical links. The theme is that for links, medial quandles provide stronger invariants than reduced Alexander modules.

Large N correlators of Chern-Simons theory in lens spaces

We compute the invariants for a class of knots and links in arbitrary representations in ${S}^{3}/{\mathbb{Z}}_{p}$ in the large $k$ (level), large $N$ (rank) limit, keeping $N/(k+N)=\ensuremath{\lambda}$ fixed, in $U(N)$ and $Sp(N)$ Chern-Simons theories. Using the relation between the saddle-point description and collective field theory, we first find that the invariants for the Hopf link and unknot are given by the on-shell collective field theory action. We next show that the results of these two invariants can be used to compute the invariants of other torus knots and links. We also discuss the large $N$ phase structure of the Hopf link invariant and observe that the same may admit a Douglas-Kazakov type phase transition depending on the choice of representations and $\ensuremath{\lambda}$.

A table of n-component handlebody links of genus n+1 up to six crossings

AbstractA handlebody link is a union of handlebodies of positive genus embedded in 3-space, which generalises the notion of links in classical knot theory. In this paper, we consider handlebody links with a genus two handlebody and $n-1$ solid tori, $n>1$ . Our main result is the classification of such handlebody links with six crossings or less, up to ambient isotopy.

Localized Boundary Knot Method for Solving Two-Dimensional Inverse Cauchy Problems

In this paper, a localized boundary knot method is adopted to solve two-dimensional inverse Cauchy problems, which are controlled by a second-order linear differential equation. The localized boundary knot method is a numerical method based on the local concept of the localization method of the fundamental solution. The approach is formed by combining the classical boundary knot method with the localization method. It has the potential to solve many complex engineering problems. Generally, in an inverse Cauchy problem, there are no boundary conditions in specific boundaries. Additionally, in order to be close to the actual engineering situation, a certain level of noise is added to the known boundary conditions to simulate the measurement error. The localized boundary knot method can be used to solve two-dimensional Cauchy problems more stably and is truly free from mesh and numerical quadrature. In this paper, the stability of the method is verified by using multi-connected domain and simply connected domain examples in Laplace equations.

Goussarov–Polyak–Viro conjecture for degree three case

Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas has been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to present the seven invariants of the degree three (Proposition 4). We further give [Formula: see text] Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak–Viro Gauss diagram formula [M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not. 1994 (1994) 445–453] is not a long virtual knot invariant; however, it is included in the list of [Formula: see text] formulas. It has been unknown whether this formula would be available by arrow diagram calculus automatically. In consequence, as it relates to the conjecture of Goussarov-Polyak-Viro [Finite-type invariants of classical and virtual knots, Topology 39 (2000) 1045–1068, Conjecture 3.C], for all the degree three finite type long virtual knot invariants, each Gauss diagram formula is represented as those of Vassiliev invariants of classical knots (Theorem 1).

H-Refinement method for toric parameterization of planar multi-sided computational domain in isogeometric analysis

Toric surface patches are a class of multi-sided surface patches that can represent multi-sided domains without mesh degeneration. In this paper, we propose an improved subdivision algorithm for toric surface patches, which subdivides an N-sided toric surface patch into N rational tensor product Bézier surface patches. By the proposed subdivision algorithm, a Ck-continuous spline surface composed of piecewise toric surface patches is subdivided into a set of rational tensor product Bézier surface patches with Gk-continuity. Additionally, after subdivision, toric surface patches are compatible with CAD systems. Combining the subdivision algorithm with the classical knot insertion algorithm of non-uniform rational B-splines, we develop a novel h-refinement scheme for isogeometric analysis with planar toric parameterizations. Several numerical examples are given to demonstrate the effectiveness and numerical stability of the presented method.

Redefining disoriented knots and diagrammatic methods

Altıntaş (2018) introduced a new concept with the name of disoriented knot. He defined a disoriented knot as an embedding of a disoriented circle with two arcs into . In this paper, we redefine a disoriented knot as an embedding of a disoriented circle with 2n arcs into and expand the diagrammatic invariants and methods of classical knot theory such as connected sum, Reidemeister moves, Gauss codes, and Gauss diagrams to disoriented knot theory. Thus, we create the basic diagrammatic invariants and methods of disoriented knot theory.

On long knots in the full torus

The aim of this paper is to realize the techniques of picture-valued invariants and invariants valued in free groups for long knots in the full torus. Such knots and links are of particular interest because of their relation to Legendrian knots, knotoids, [Formula: see text]-manifolds and many other objects. Invariants constructed in the paper are powerful and easy to compare. This paper is a sequel of [V. O. Manturov, A free-group valued invariant of free knots, preprint (2020), arXiv:2012.15571v2]. Long knots naturally appear in the study of classical knots [T. Fiedler, More [Formula: see text]-cocycles for classical knots, preprint (2020), arXiv:2004.04624; A. Mortier, A Kontsevich integral of order [Formula: see text], preprint (2018), arXiv:1810.05747].

Polynomial Generalizations of Knot Colorings

In the field of knot theory, knot invariants are properties preserved across all embeddings and projections of the same knot. Fox n-coloring is a classical knot invariant which associates to each knot projection a system of linear equations. We generalize Fox’s n-coloring by using two, not necessarily distinct, polynomials over a field F, which we say form a (g,f)F coloring. We introduce a sufficient condition, called strong, for a pair of polynomials to form a (g,f)F coloring. We confirm a family of pairs of linear polynomials each of which form a (g,f)F coloring. We prove that there are no strong pairs containing an irreducible quadratic polynomial over a field F not of characteristic two. Furthermore, we find a method to produce polynomials with unbounded degree that form colorings over suitable fields.

A practical method for computing with piecewise Chebyshevian splines

A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion. The interest in such kind of spaces is justified by the fact that, similarly as for polynomial splines, the related parametric curves exhibit the desired properties of convex hull inclusion, variation diminution and intuitive relation between the curve shape and the location of the control points. For a good-for-design space, in this paper we construct a set of functions, called transition functions, which allow for efficient computation of the B-spline basis, even in the case of nonuniform and multiple knots. Moreover, we show how the spline coefficients of the representations associated with a refined knot partition and with a raised order can conveniently be expressed by means of transition functions. This result allows us to provide effective procedures that generalize the classical knot insertion and degree raising algorithms for polynomial splines. We further discuss how the approach can straightforwardly be generalized to deal with geometrically continuous piecewise Chebyshevian splines as well as with splines having section spaces of different dimensions. From a numerical point of view, we show that the proposed evaluation method is easier to implement and has higher accuracy than other existing algorithms.

Cocycle Invariants and Oriented Singular Knots

We extend the quandle cocycle invariant to oriented singular knots and links using algebraic structures called oriented singquandles and assigning weight functions at both regular and singular crossings. This invariant coincides with the classical cocycle invariant for classical knots, but provides extra information about singular knots and links. The new invariant distinguishes the singular granny knot from the singular square knot.

Finite type invariants for knotoids

We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the Vassiliev skein relation. Then, for spherical knotoids we show that there are non-trivial type-1 invariants, in contrast with classical knot theory where type-1 invariants vanish. We give a complete theory of type-1 invariants for spherical knotoids, by classifying linear chord diagrams of order one, and we present examples arising from the affine index polynomial and the extended bracket polynomial.

Chaos and integrability in -geometry

We review the integrability of the geodesic flow on a threefold admitting one of the three group geometries in Thurston’s sense. We focus on the case. The main examples are the quotients , where is a cofinite Fuchsian group. We show that the corresponding phase space contains two open regions with integrable and chaotic behaviour, with zero and positive topological entropy, respectively. As a concrete example we consider the case of the modular threefold with the modular group . In this case is known to be homeomorphic to the complement of a trefoil knot in a 3-sphere. Ghys proved the remarkable fact that the lift of a periodic geodesic on the modular surface to produces the same isotopy class of knots as that which appears in the chaotic version of the celebrated Lorenz system and was studied in detail by Birman and Williams. We show that these knots are replaced by trefoil knot cables in the integrable limit of the geodesic system on . Bibliography: 60 titles.

Concordance Invariants and the Turaev Genus

Abstract We show that the differences between various concordance invariants of knots, including Rasmussen’s $s$-invariant and its generalizations $s_n$-invariants, give lower bounds to the Turaev genus of knots. Using the fact that our bounds are nontrivial for some quasi-alternating knots, we show the additivity of Turaev genus for a certain class of knots. This leads us to the 1st example of an infinite family of quasi-alternating knots with Turaev genus exactly $g$ for any fixed positive integer $g$, solving a question of Champanerkar–Kofman.

A Homological Casson Type Invariant of Knotoids

We consider an analogue of the well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. The value of the extension is a formal sum of subgroups of the first homology group $$H_1(\varSigma )$$ where $$\varSigma $$ is an oriented surface with (maybe) non-empty boundary in which knotoid diagrams lie. To make the extension informative for spherical knotoids it is sufficient to transform an initial knotoid diagram in $$S^2$$ into a knotoid diagram in the annulus by removing small disks around its endpoints. As an application of the invariants we prove two theorems: a sharp lower bound of the crossing number of a knotoid (the estimate differs from its prototype for classical knots proved by M. Polyak and O. Viro in 2001) and a sufficient condition for a knotoid in $$S^2$$ to be a proper knotoid (or pure knotoid with respect to Turaev’s terminology). Finally we give a table containing values of our invariants computed for all spherical prime proper knotoids having diagrams with at most 5 crossings.

A family of polynomial invariants for virtual knots

As a generalization of the classical knots, virtual knots deal with the problem of circles embedded in a thickened orientable surface with arbitrary genus. In the study of virtual knots, the classification of virtual knots remains an important issue. In recent years, many polynomial invariants have appeared, such as affine index polynomial, a new invariant RD[t](h) of virtual knots, etc. In this paper, we introduce a family of new polynomial invariants H-polynomial HDn(t,h,ℓ)(n∈N⁎) of virtual knots, and we construct a series of examples of virtual knots which have the same RD[t](h) but different HDn(t,h,ℓ) for some n. Finally, we get the relationship between HDn(t,h,ℓ) and HD⁎n(t,h,ℓ), where D⁎ is obtained from D by interchanging over and under lines at the classical crossings.

Unknotting with a single twist

Given a knot K \subset S^3 , is it possible to unknot it by performing a single twist, and if so, what are the possible linking numbers of such a twist? We develop obstructions to unknotting using a twist of a specified linking number. The obstructions we describe are built using classical knot invariants, Casson–Gordon invariants, and Heegaard Floer theory.

A relation between the crossing number and the height of a knotoid

Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such arcs which are disjoint from crossings. In the paper, we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.

More 1-cocycles for classical knots

Let [Formula: see text] be the topological moduli space of long knots up to regular isotopy, and for any natural number [Formula: see text] let [Formula: see text] be the moduli space of all [Formula: see text]-cables of framed long knots which are twisted by a string link to a knot in the solid torus [Formula: see text]. We upgrade the Vassiliev invariant [Formula: see text] of a knot to an integer valued combinatorial 1-cocycle for [Formula: see text] by a very simple formula. This 1-cocycle depends on a natural number [Formula: see text] with [Formula: see text] as a parameter and we obtain a polynomial-valued 1-cocycle by taking the Lagrange interpolation polynomial with respect to the parameter. We show that it induces a non-trivial pairing on [Formula: see text] already for [Formula: see text].