Journal of Knot Theory and Its Ramifications

It is known that there exists a surjective map from the set of weak (1, 3) homotopy classes of knot projections to the set of positive knots [N. Ito and Y. Takimura, (1, 2) and weak (1, 3) homotopies on knot projections, J. Knot Theory Ramifications 22 (2013) 1350085]. An interesting question whether this map is also injective, which question was formulated independently by S. Kamada and Y. Nakanishi in 2013 (Question q1). This paper obtains an answer to this question.

A finitely generated group $G$ acting on a tree with infinite cyclic edge and vertex stabilizers is called a generalized Baumslag–Solitar group (GBS group). We prove that a one-knot group $G$ is a GBS group if and only if $G$ is a torus knot group, and describe all n-knot GBS groups for $n\ge 3$.

In [V. O. Manturov, An almost classification of free knots, Dokl. Math. 88(2) (2013) 556–558.] the second author constructed an invariant which in some sense generalizes the quantum $\mathrm{\text{sl}}(3)$ link invariant of Kuperberg to the case of free links. In this paper, we generalize this construction to free graph-links. As a result, we obtain an invariant of free graph-links with values in linear combinations of graphs. The main property of this invariant is that under certain conditions on the representative of the free graph-link, we can recover this representative from the value invariant on it. In addition, this invariant allows one to partially classify free graph-links.

In the 1950s Milnor defined a family of higher-order invariants generalizing the linking number. Even the first of these new invariants, the triple linking number, has received fruitful study since its inception. In the case that a link $L$ has vanishing pairwise linking numbers, this triple linking number gives an integer-valued invariant. When the linking numbers fail to vanish, the triple linking number is only well-defined modulo their greatest common divisor. In recent work Davis–Nagel–Orson–Powell produce a single invariant called the total triple linking number refining the triple linking number and taking values in an abelian group called the total Milnor quotient. They present examples for which this quotient is nontrivial even though each of the individual triple linking numbers take values in the trivial group, $\mathbb{Z}/1$, and so carry no information. As a consequence, the total triple linking number carries more information than do the classical triple linking numbers. The goal of this paper is to compute this group and show that when $L$ is a link of at least six components it is nontrivial. Thus, this total triple linking number carries information for every $(n\ge 6)$-component link, even though the classical triple linking numbers often carry no information.

We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by terms of the lower central series of knot groups. Also, we study decomposition of virtual knot groups as semi direct product and free product with amalgamation. In particular, we prove that the groups of some virtual knots are extensions of finitely generated free groups by infinite cyclic groups.

In 2003, Ozsváth and Szabó defined the concordance invariant $\tau$ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of $\tau$ for knots in $S^3$ and a combinatorial proof that $\tau$ gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in $S^3$, extending HFK for knots. We define a $\mathbb{Z}$-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined $\tau$ invariant for balanced spatial graphs generalizing the $\tau$ knot concordance invariant. In particular, this defines a $\tau$ invariant for links in $S^3$. Using techniques similar to those of Sarkar, we show that our $\tau$ invariant is an obstruction to a link being slice.

Ng constructed an invariant of knots in ${ℝ}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${ℝ}^4$ using marked graph diagrams.

A twisted torus knot is a torus knot with some consecutive strands twisted. More precisely, a twisted torus knot $T(p,q,r,s)$ is a torus knot $T(p,q)$ with $r$ consecutive strands $s$ times fully twisted. We determine which twisted torus knots $T(p,q,p-kq,-1)$ are a torus knot.

For any classical knot $k^1$, we can construct a ribbon $2$-knot $\mathrm{\text{spun}}(k^1)$ by spinning an arc removed a small segment from $k^1$ about $R^2$ in $R^4$. A ribbon $2$-knot is an embedded $2$-sphere in $R^4$. If $k^1$ has an $n$-crossing presentation, by spinning this, we can naturally construct a ribbon presentation with $n$ ribbon crossings for $\mathrm{\text{spun}}(k^1)$. Thus, we can define naturally a notion on ribbon $2$-knots corresponding to the crossing number on classical knots. It is called the ribbon crossing number. On classical knots, it was a long-standing conjecture that any odd crossing classical knot is not amphicheiral. In this paper, we show that for any odd integer $n$ there exists an amphicheiral ribbon $2$-knot with the ribbon crossing number $n$.