Related papers: Knot modules and ribbon 2-knots

We prove that hyperbolic 2-bridge knots are determined amongst all compact 3-manifolds by the profinite completions of their knot groups.

A torti-rational knot, denoted by K(2a,b|r), is a knot obtained from the 2-bridge link B(2a,b) by applying Dehn twists an arbitrary number of times, r, along one component of B(2a,b). We determine the genus of K(2a,b|r) and solve a question…

Given a positive integer $n$, we say that two knots are $V_n$-equivalent if they have the same Vassiliev invariants of order $\le n$. We showed that the $V_n$-equivalence classes of ribbon knots form a group, the operation being induced by…

We exhibit infinitely many ribbon knots, each of which bounds infinitely many pairwise non-isotopic ribbon disks whose exteriors are diffeomorphic. This family provides a positive answer to a stronger version of an old question of Hitt and…

We consider ribbon n-knots for n\geq 2. For such knots we define a set of moves on ribbon disks, and show that any two ribbon disks for isotopic knots are related by a finite sequence of such moves and ambient isotopies. Using this we are…

Let $S$ be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if $R$ is a non-Gorenstein quotient of $S$ of small colength, then every totally reflexive $R$-module is free. Indeed, the second syzygy of the…

We prove that for a countable, commutative ring $R$, the class of countable $R$-modules either has only countably many isomorphism types, or else it is Borel complete. The machinery gives a succinct proof of the Borel completeness of TFAB,…

We study knots of order 2 in the grope filtration $\{\G_h\}$ and the solvable filtration $\{\F_h\}$ of the knot concordance group. We show that, for any integer $n\ge4$, there are knots generating a $\Z_2^\infty$ subgroup of…

We show that for many classical knots one can find generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the…

Let T denote the group of smooth concordance classes of topologically sice knots. We show that the first quotient in the bipolar filtration of T (i.e. 0-bipolar knots modulo 1-bipolar knots) has infinite rank, even modulo Alexander…

Let $\mathcal{T}$ be the group of smooth concordance classes of topologically slice knots, and $\{0\}\subset\cdots\subset \mathcal{T}_{n+1}\subset\mathcal{T}_{n}\subset \cdots\subset \mathcal{T}_{0}\subset \mathcal{T}$ be the bipolar…

In a group, a non-trivial element is called a generalized torsion element if some non-empty finite product of its conjugates equals to the identity. We say that a knot has generalized torsion if its knot group admits such an element. For a…

We classify fibered ribbon pretzel knots up to mutation. The classification is complete, except perhaps for members of Lecuona's ``exceptional'' family of [Lec15]. The result is obtained by combining lattice embedding techniques with…

We discuss topological objects, in particular the non-Abrikosov vortex and the magnetic knot made of the twisted non-Abrikosov vortex, in two-gap superconductor. We show that there are two types of non-Abrikosov vortex in Ginzburg-Landau…

Let X' be the toroidal compactification of the quotient of the complex 2-ball by a torsion free lattice G of SU(2,1). We say that X'is co-abelian if there is an abelian surface, birational to X'. The present work can be viewed as an…

We show that the braided tensor category of finitely-generated weight modules for the simple affine vertex operator algebra $L_k(\mathfrak{sl}_2)$ of $\mathfrak{sl}_2$ at any admissible level $k$ is rigid and hence a braided ribbon…

We establish a bijection between torsion pairs in the category of finite-dimensional modules over a finite-dimensional algebra A and pairs (Z, I) formed by a closed rigid set Z in the Ziegler spectrum of A and a set I of indecomposable…

A conjecture of Shumakovitch states that every nontrivial knot has 2-torsion in its Khovanov homology. We show that if a knot $K$ has no 2-torsion in its Khovanov homology, then the rank of its reduced Khovanov homology is minimal among all…

It is well known that there are many 2-torsion elements in the classical knot concordance group. On the other hand, it is not known if there is any torsion element in the rational knot concordance group $\mathcal{C}_\mathbb{Q}$. Cha defined…

We present a new theory which describes the collection of all tunnels of tunnel number 1 knots in the 3-sphere (up to orientation-preserving equivalence in the sense of Heegaard splittings) using the disk complex of the genus-2 handlebody…