Journal of Knot Theory and Its Ramifications

Quandle cocycle invariants form a powerful and well-developed tool in knot theory. This paper treats their variations — namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow world are extended to the positive and twisted cases. Another application is a sharpening of twisted quandle cocycle invariants for multi-component links.

A ribbon chord diagram, or simply a chord diagram, of a ribbon surface-link in the 4-space is introduced. Links, virtual links and welded virtual links can be described naturally by chord diagrams with the corresponding moves, respectively. Some moves on chord diagrams are introduced by overseeing these special moves. Then the faithful equivalence on ribbon surface-links is stated in terms of the moves on chord diagrams. This answers questions by Nakanishi and Marumoto affirmatively. The faithful TOP-equivalence on ribbon surface-links derives the same result. By combining a previous result on TOP-triviality of a surface-knot, a ribbon surface-knot is DIFF-trivial if and only if the fundamental group is an infinite cyclic group. This corrects an erroneous proof in Yanagawa's old paper.

A. S. Lipson constructed two state models yielding the same classical link invariant obtained from the Kauffman polynomial F(a, u). In this paper, we apply Lipson's state models to marked graph diagrams of surface-links, and observe when they induce surface-link invariants.

We present a new method to produce simple formulas for 1-cocycles of knots over the integers, inspired by Polyak–Viro's formulas for finite-type knot invariants. We conjecture that these 1-cocycles represent finite-type cohomology classes in the sense of Vassiliev. An example of degree 3 is studied, and shown to coincide over ℤ₂ with the Teiblum–Turchin cocycle .

We introduce the side parameter map from the space of cone hyperbolic structures on the torus with a single cone point to the hyperbolic plane ℍ², based on the idea in the Jorgensen theory for punctured torus Kleinian groups. Then, we show that the side parameter map is a homeomorphism.

Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words was introduced by Turaev [Topology of words, Proc. Lond. Math. Soc.95(3) (2007) 360–412], who thought all free knots to be trivial). As it turned out, these new objects are highly nontrivial, see [V. O. Manturov, Parity in knot theory, Mat. Sb.201(5) (2010) 65–110], and even admit nontrivial cobordism classes [V. O. Manturov, Parity and cobordisms of free knots, Mat. Sb.203(2) (2012) 45–76]. An important issue is the existence of invariants where a diagram evaluates to itself which makes such objects "similar" to free groups: An element has its minimal representative which "lives inside" any representative equivalent to it. In this paper, we consider generalizations of free knots by means of (finitely presented) groups. These new objects have lots of nontrivial properties coming from both knot theory and group theory. This connection allows one not only to apply group theory to various problems in knot theory but also to apply Reidemeister moves to the study of (finitely presented) groups. Groups appear naturally in this setting when graphs are embedded in surfaces.

In previous papers (see e.g. [V. O. Manturov, Parity in knot theory, Mat. Sbornik201(5) (2010) 65–110]), the author realized the following principle for many knot theories: if a knot diagram is complicated enough then it reproduces itself, i.e. a subdiagram of any other diagram equivalent to it. This principle is realized by diagram-valued invariants [⋅] of knots such that [K] = K for K complicated enough. It turns out that in the case of free braids, the same principle can be realized in an unexpectedly easy way by a one-term invariant formula.

In this paper, we study the Gauss diagrams for periodic virtual knots (Theorem 3.1) and show that the virtual knot corresponding to a periodic Gauss diagram is equivalent to the periodic virtual knot whose factor is the virtual knot corresponding to the factor Gauss diagram (Theorem 3.2). We give formulae for the writhe polynomial and the affine index polynomial of periodic virtual knots by using those of factor knots (Corollary 4.2, Corollary 4.6).

An unknotting operation is a local move on a knot diagram such that any knot diagram can be transformed into a diagram of the unknot by a finite sequence of the operations and Reidemeister moves. In this paper, we introduce a new local move H(T) on a knot diagram which is obtained by the rotation of a tangle diagram T and study their properties. As an application, we prove that the H(T)-move is an unknotting operation for any descending tangle diagram T.

By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken surface diagrams. A marked graph diagram is a link diagram possibly with 4-valent vertices equipped with markers. Lomonaco, Jr. and Yoshikawa introduced a method of describing surface-links by using marked graph diagrams. In this paper, we give interpretations of these quandle cocycle invariants in terms of marked graph diagrams, and introduce a method of computing them from marked graph diagrams.

Let be a hyperbolic transformation. Let B be a new band attaching to L such that is also a hyperbolic transformation. In this paper, we will study the relationship between the realizing surfaces and . If B is a noncoherent band to both L and such that is defined, then and are ambient isotopic, where RP² is one of the standard real projective planes. We will study the triviality of because as an application, RP² can untangle some knotted sphere with suitable conditions, when it is attached to by the connected sum.

We list prime knots with up to 12 crossings having the property: the lower bound of the unknotting number cannot be decided by the signature or non-triviality but is given by either: (i) the condition of a 2-bridge knot with unknotting number one, (ii) the criteria using special values of the Jones, Q, or HOMFLYPT polynomials, or (iii) Wendt's formula. Then we can give new information to the table of unknotting numbers in "KnotInfo", a web-based table of knot invariants.

The oriented Gordian distance between two oriented links is the minimal number of crossing changes needed to deform one into the other. We compile a table of oriented Gordian distances between 2-component non-splittable links with up to six crossings. In particular, we give a criterion of oriented Gordian distance two using a special value of the Jones polynomial, which allows us to prove that the unlinking number of the 2-component link is 3. This is one of the five links for which Kohn could not compute the unlinking number (recently, Nagel and Owens [12] have settled the unlinking numbers of these links).