Journal of Knot Theory and Its Ramifications

The stabilization height of a fiber surface in the 3-sphere is the minimal number of Hopf plumbing operations needed to attain a stable fiber surface from the initial surface. We show that families of fiber surfaces related by iterated Stallings twists have unbounded stabilization height.

The problem of faithfulness of the (reduced) Burau representation for $n=4$ is known to be equivalent to the problem of whether certain two matrices $A$ and $B$ generate a free group of rank two. In this note, we give a simple proof that $〈A^3,B^3〉$ is a free group of rank two.

We define and calculate signature and nullity invariants for complex schemes for curves in $ℝP(2)$. We use an analog of the Murasugi–Tristram inequality to prohibit certain schemes from being realized by real algebraic curves. We give new formulas for Casson–Gordon invariants of graph manifolds, and signatures of graph links.

We compute the Dubrovnik skein module of the lens spaces $L_{p,1}$, $p>1$, as well as the Kauffman two variables skein module when $p$ is odd. We also show that there is torsion in the Kauffman skein module when $p$ is even.

In this paper, we construct a new homology theory for semi-groups satisfying the right self-distributivity axiom or the idempotency axiom. Next, we consider the geometric realization corresponding to the homology theory. We continue with the comparison of this homology theory with one-term and two-term (rack) homology theories of self-distributive algebraic structures. Finally, we propose connections between the homology theory and knot theory via Temperley–Lieb algebras.

We work with a generalization of knot theory, in which one diagram is reachable from another via a finite sequence of moves if a fixed condition regarding the existence of certain morphisms in an associated category is satisfied for every move of the sequence. This conditional setting leads to a possibility of irreversible moves, terminal states, and to using functors more general than the ones used as knot invariants. Our main focus is the category of diagrams with a binary relation on the set of arcs, indicating which arc can move over another arc. We define homology of binary relations, and merge it with quandle homology, to obtain the homology for partial quandles with a binary relation. This last homology can be used to analyze link diagrams with a binary relation on the set of components.

Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. We discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme–Guizon, Przytycki and Rong. An explicit formula for the rank of the third chromatic homology group on the main diagonal is given and used to compute the corresponding Khovanov homology group and the fourth coefficient of the Jones polynomial for links with certain diagrams.

A natural topology on the set of left orderings on free abelian groups and free groups $F_n$, $n>1$ has studied in [A. S. Sikora, Topology on the spaces of orderings of groups, Bull. London Math. Soc. 36(4) (2004) 519–526; L. Smith, On ordering free groups, J. Symbolic Comput. 40 (2005) 1285–1290, Corrigendum (with A. Clay) 44 (2009) 1529–1532]. It has been proven already that in the abelian case the resulted topological space is a Cantor set. There was a conjecture: this is also true for the free group $F_n$ with $n>1$ generators. We point out the paper dealing with equivalent questions.

We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 crossings. Following an approach of Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed their Jones polynomials by a divide-and-conquer method, and tested those with trivial Jones polynomials for unknottedness with the computer program SnapPy. We employed numerous novel strategies for reducing the computation time per knot diagram and the number of knot diagrams to be considered. That made computations up to 21 crossings possible on a single processor desktop computer. We explain these strategies in this paper. We also provide total numbers of algebraic tangles up to 18 crossings and of Conway polyhedra up to 22 vertices. We encountered new unknot diagrams with no crossing-reducing pass moves in our search. We report one such diagram in this paper.

We discuss the higher order stabilization of the coefficients of the colored Jones polynomial. In particular, we find an expression for the second stable sequence of the colored Jones polynomial of a certain class of knots. We also determine which knots have the same higher order stability.

Let $G$ be a finite group acting orthogonally on a pair $(S^d,\mathrm{\Gamma })$ where $\mathrm{\Gamma }$ is a finite, connected graph of genus $g>1$ embedded in the sphere $S^d$. The 3-dimensional case $d=3$ has recently been considered in a paper by C. Wang, S. Wang, Y. Zhang and the present author where for each genus $g>1$, the maximum order of an orientation-preserving $G$-action on a pair $(S^3,\mathrm{\Gamma })$ is determined and the corresponding graphs $\mathrm{\Gamma }$ are classified (an upper bound for the order of $G$ is $12(g-1)$). In the present paper, we consider arbitrary dimensions $d$ and prove that the order of $G$ is bounded above by a polynomial of degree $d/2$ in $g$ if $d$ is even and of degree $(d+1)/2$ if $d$ is odd; moreover, the degree $d/2$ is best possible in even dimensions $d$. We discuss also the problem, given a finite graph $\mathrm{\Gamma }$ and its finite symmetry group, to find the minimal dimension of a sphere into which $\mathrm{\Gamma }$ embeds equivariantly as above.

This paper refers to the work [V. Goryunov, Local invariants of maps between 3-manifolds, J. Topology 6 (2013) 757–776] on local invariants of maps between 3-manifolds. It is assumed that the manifolds have no boundary, and that the source is compact. In the case when the source and the target are oriented, Goryunov proved that every local order one invariant with integer values can be written as a linear combination of seven basic invariants, and gave a geometrical interpretation for them. When the target is the oriented ${ℝ}^3$, there are further four basic mod2 invariants. One of the mod2 invariants has been provided with a topological interpretation, in terms of the number of components and of the self-linking of a framed link constructed from the cuspidal edge. Here, we show that two further independent linear combinations of the mod2 invariants have a topological interpretation, involving the self-linking number of two curves defined by all irregular points of the critical value set of a generic map from an oriented closed 3-manifold to ${ℝ}^3$.