Related papers: Distinguishing 2-knots admitting circle actions by…

A branched twist spin is a generalization of twist spun knots, which appeared in the study of locally smooth circle actions on the $4$-sphere due to Montgomery, Yang, Fintushel and Pao. In this paper, we give a sufficient condition to…

The union of singular orbits of an effective locally smooth circle action on the 4-sphere consists of two 2-knots, $K$ and $K^{\prime}$, intersecting at two points transversely. Each of $K$ and $K^{\prime}$ is called a branched twist spin.…

It is known that a presentation of the knot group of a branched twist spin is obtained from a Wirtinger presentation of the original 1-knot group by adding a generator corresponding to a regular orbit of the circle action and a certain…

We characterize the groups of branched twist spins of classical knots in terms of 3-manifold groups, and also give a purely algebraic, conjectural characterization in terms of $PD_3$-groups. We show also that each group is the group of at…

For any integer n > 2, the n-fold cyclic branched cover M of an alternating prime knot K in the 3-sphere determines K, meaning that if K is a knot in the 3-sphere that is not equivalent to K then its n-fold cyclic branched cover cannot be…

The trace of $n$-framed surgery on a knot in $S^3$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded 2-sphere…

We give uniform, explicit, and simple face-pairing descriptions of all the branched cyclic covers of the 3-sphere, branched over two-bridge knots. Our method is to use the bi-twisted face-pairing constructions of Cannon, Floyd, and Parry;…

We give a first example of 2-knots with the same knot group but different knot quandles by analyzing the knot quandles of twist spins. As a byproduct of the analysis, we also give a classification of all twist spins with finite knot…

We show that, for any prime p, a knot K in the 3-sphere is determined by its p-fold cyclic unbranched covering. We also investigate when the m-fold cyclic unbranched covering of a knot coincides with the n-fold cyclic unbranched covering of…

Some generalizations and variations of the Fintushel-Stern rim surgery are known to produce smoothly knotted surfaces. We show that if the fundamental groups of their complements are cyclic, then these surfaces are topologically unknotted.…

We discuss 3-manifolds which are cyclic coverings of the 3-sphere, branched over 2-bridge knots and links. Different descriptions of these manifolds are presented: polyhedral, Heegaard diagram, Dehn surgery and coloured graph constructions.…

A knot in the 3-sphere is called doubly slice if it is a slice of an unknotted 2-sphere in the 4-sphere. We give a bi-sequence of new obstructions for a knot being doubly slice. We construct it following the idea of Cochran-Orr-Teichner's…

For some families of two-bridge knots, including double-twist knots with genus at least four, we determine precisely the set of integers $n>1$ such that the fundamental group of the $n$-fold cyclic branched cover of the 3-sphere along these…

Many three dimensional manifolds are two-fold branched covers of the three dimensional sphere. However, there are some that are not. This paper includes exposition about two-fold branched covers and many examples. It shows that there are…

For each integer $d$ at least two, we construct non-spin closed oriented flat manifolds with holonomy group $\mathbb Z_2^d$ and with the property that all of their finite proper covers have a spin structure. Moreover, all such covers have…

Using spinning we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere. Consider the 2-torus standardly embedded in the 3-sphere, which is further standardly embedded in the 6-sphere. At each point of the…

Updated rerefences and introduction. Given a knot in an integer homology sphere, one can construct a family of closed 3-manifolds (parametrized by the positive integers), namely the cyclic branched coverings of the knot. In this paper we…

We show that any finitely presented group with an index two subgroup is realized as the fundamental group of a closed smooth non-orientable four-manifold that admits an exotic smooth structure, which is obtained by performing a Gluck twist.…

For any alternating knot, it is known that the double branched cover of the $3$-sphere branched over the knot is an $L$-space. We show that the three-fold cyclic branched cover is also an $L$-space for any genus one alternating knot.

Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…