Related papers: Representing knots by filling Dehn spheres
We introduce the notion of round surgery diagrams in $S^3$ for representing 3-manifolds similar to Dehn surgery diagrams. We give a correspondence between a certain class of round surgery diagrams and Dehn surgery diagrams for 3-manifolds.…
A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated…
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…
We generalize the results of [AS], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each the lift of a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a…
Two Dehn surgeries on a knot are called {\it purely cosmetic}, if they yield manifolds that are homeomorphic as oriented manifolds. Suppose there exist purely cosmetic surgeries on a knot in $S^3$, we show that the two surgery slopes must…
We prove that every cusped hyperbolic 3-manifold has a finite cover admitting infinitely many geometric ideal triangulations. Furthermore, every long Dehn filling of one cusp in this cover admits infinitely many geometric ideal…
The cosmetic crossing conjecture (also known as the "nugatory crossing conjecture") asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery…
We apply representation theory to study the homology of equivariant Dehn-fillings of a given finite, regular cover of a compact 3-manifold with boundary a torus. This yields a polynomial which gives the rank of the part of the homology…
We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc…
We study the geometry of hyperbolic knots that admit alternating projections on embedded surfaces in closed 3-manifolds. We show that, under mild hypothesis, their cusp area admits two sided bounds in terms of the twist number of the…
Examples of knots and links distinguished by the total rank of their Khovanov homology but sharing the same two-fold branched cover are given. As a result, Khovanov homology does not yield an invariant of two-fold branched covers.
Given a 3-manifold $Y$ and a free homotopy class in $[S^1,Y]$, we investigate the set of topological concordance classes of knots in $Y \times [0,1]$ representing the given homotopy class. The concordance group of knots in the 3-sphere acts…
We use Heegaard splittings to give a criterion for a tunnel number one knot manifold to be non-fibered and to have large cyclic covers. We also show that such a knot manifold (satisfying the criterion) admits infinitely many virtually Haken…
We present a practical algorithm to test whether a 3-manifold given by a triangulation or an ideal triangulation contains a closed essential surface. This property has important theoretical and algorithmic consequences. As a testament to…
We show that an immersed thrice-punctured sphere in a cusped orientable hyperbolic 3-manifold is either embedded or has a single clasp in a manifold obtained by hyperbolic Dehn filling on a cusp of the Whitehead link complement.
A combinatorial presentation of closed orientable 3-manifolds as bi-tricolored links is given together with two versions of a calculus via moves to manipulate bi-tricolored links without changing the represented manifold. That is, we…
Given a real analytic function $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated critical point at the origin, the link $L_f$ of the singularity is a real fibred knot in $\mathbb{S}^{3}$. From this singularities, we construct a family…
We show that any homologically non-trivial Dehn twist of a compact surface F with boundary is the lifting of a half-twist in the braid group B_n, with respect to a suitable branched covering p : F -> B^2. In particular, we allow the surface…
A knotted surface in the 4-sphere may be described by means of a hyperbolic diagram that captures the 0-section of a special Morse function, called a hyperbolic decomposition. We show that every hyperbolic decomposition of a knotted surface…
We are interested in finite groups acting orientation-preservingly on 3-manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain an involution with nonempty connected fixed point…