Journal of Knot Theory and Its Ramifications

Study of equivalence classes of links up to n-moves plays an important role in the theory of invariants based on the skein relation and, in particular, skein modules. In this paper, we consider Nakanishi's 4-move conjecture [12]. The modification of the conjecture to 2-component link (homotopically trivial links) is a question proposed by Kawauchi [10]. We define a new invariant of links which is preserved by 4-moves and analyze its potential strength. In particular, we show that our invariant allows us to obtain results of [8, 9, 13] concerning 4-moves.

We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group. We also conjecture a characterization of knots whose invariants of all orders vanish in terms of their Seifert surfaces.

Computing unlinking number is usually very difficult and complex problem, therefore we define BJ-unlinking number and recall Bernhard–Jablan conjecture stating that the classical unknotting/unlinking number is equal to the BJ-unlinking number. We compute BJ-unlinking number for various families of knots and links for which the unlinking number is unknown. Furthermore, we define BJ-unlinking gap and construct examples of links with arbitrarily large BJ-unlinking gap. Experimental results for BJ-unlinking gap of rational links up to 16 crossings, and all alternating links up to 12 crossings are obtained using programs LinKnot and K2K. Moreover, we propose families of rational links with arbitrarily large BJ-unlinking gap and polyhedral links with constant non-trivial BJ-unlinking gap. Computational results suggest existence of families of non-alternating links with arbitrarily large BJ-unlinking gap.

We give two proofs that the 3-torus is not weakly d-congruent to #³S¹ × S², if d > 2. We study how cohomology ring structure relates to weak congruence. We give an example of three 3-manifolds which are weakly 5-congruent but are not 5-congruent.

This paper studies the biquandle as an invariant of virtual knots and links, analyzing its structure via composition of biquandle morphisms where elementary morphisms are associated with the crossings and virtual crossings in the diagram.

From a pseudo-triangulation with n tetrahedra T of an arbitrary closed orientable connected 3-manifold (for short, a 3D-space) M³, we present a gem J′, inducing 𝕊³, with the following characteristics: (a) its number of vertices is O(n); (b) it has a set of p pairwise disjoint couples of vertices {u_i, v_i}, each named a twistor; (c) in the dual (J′)^⋆ of J′ a twistor becomes a pair of tetrahedra with an opposite pair of edges in common, and it is named a hinge; (d) in any embedding of (J′)^⋆ ⊂ 𝕊³, the ∊-neighborhood of each hinge is a solid torus; (e) these p solid tori are pairwise disjoint; (f) each twistor contains the precise description on how to perform a specific surgery based in a Denh–Lickorish twist on the solid torus corresponding to it; (g) performing all these p surgeries (at the level of the dual gems) we produce a gem G′ with |G′| = M³; (h) in G′ each such surgery is accomplished by the interchange of a pair of neighbors in each pair of vertices: in particular, |V(G′) = |V(J′)|.

This is a new proof, based on a linear polynomial algorithm, of the classical theorem of Wallace (1960) and Lickorish (1962) that every 3D-space has a framed link presentation in 𝕊³ and opens the way for an algorithmic method to actually obtaining the link by an O(n²)-algorithm. Actually this has been done and awaits a proper implentation.

It is known that the number of biquandle colorings of a long virtual knot diagram, with a fixed color of the initial arc, is a knot invariant. In this paper, we construct a more subtle invariant: a family of biquandle endomorphisms obtained from the set of colorings and longitudinal information.

We use recently introduced Rasmussen invariant to find knots that are topologically locally-flatly slice but not smoothly slice. We note that this invariant can be used to give a combinatorial proof of the slice-Bennequin inequality. Finally, we compute the Rasmussen invariant for quasipositive knots and show that most of our examples of non-slice knots are not quasipositive and, to the best of our knowledge, were previously unknown.

We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main tools are Jones and Kauffman polynomials and the fundamental group of the 2-fold branch cover of S³ along a link. We use also the fact that a 5-move is a composition of two rational ±(2, 2)-moves (i.e. -moves) and rational moves can be analyzed using the group of Fox colorings and its non-abelian version, the Burnside group of a link. One curious observation is that links related by one (2, 2)-move are not 5-move equivalent. In particular, we partially classify (up to 5-moves) 3-braids, pretzel and Montesinos links, and links up to 9 crossings.

This paper studies a non-linear model for the damped oscillations of a pendulum.

Given a knot theory (virtual, singular, knots in a 3-manifold etc.), there are deep relations between the diagrammatic knot equivalence in this theory, the braid structures and a corresponding braid equivalence. The L-moves between braids, due to their fundamental nature, may be adapted to any diagrammatic situation in order to formulate a corresponding braid equivalence. In this short paper, we discuss and compare various diagrammatic set-ups and results therein, in order to draw the underlying logic relating diagrammatic isotopy, braid structures, Markov theorems and L-move analogues. Finally, we apply our conclusions to singular braids.