Related papers: Multiple bridge surfaces restrict knot distance

The cobordism distance between knots, d(K,J), equals the four-genus g_4(K # -J). We consider d(K,K^r), where K^r is the reverse of K. It is elementary that 0 \le d(K,K^r) \le 2g_4(K) and it is known that there are knots K for which d(K,K^r)…

A slope $p/q$ is characterising for a knot $K \subset \mathbb{S}^3$ if the orientation-preserving homeomorphism type of the manifold $\mathbb{S}^3_K(p/q)$ obtained by performing Dehn surgery of slope $p/q$ along $K$ uniquely determines the…

For the purposes of this paper, Dehn surgery along a curve K in a 3-manifold M with slope r is `exceptional' if the resulting 3-manifold M_K(r) is reducible or a solid torus, or the core of the surgery solid torus has finite order in the…

Kuperberg [Algebr. Geom. Topol. 3 (2003) 587-591] has shown that a virtual knot corresponds (up to generalized Reidemeister moves) to a unique embedding in a thichened surface of minimal genus. If a virtual knot diagram is equivalent to a…

Given any connected, open 3-manifold $U$ having finitely many ends, a non-compact 3-manifold $M$ is constructed having the following properties: the interior of $M$ is homeomorphic to $U$; the boundary of $M$ is the disjoint union of…

Understanding ideal points in the character varieties of knot complements has led to a number of important invariants for 3-manifolds. Ohtsuki (1994) counted the ideal points for character varieties of 2-bridge knot complements, and he made…

We give a simple combinatoric proof of an exponential upper bound on the number of distinct 3-manifolds that can be constructed by successively identifying nearest neighbour pairs of triangles in the boundary of a simplicial 3-ball and show…

We show that if $p/q$-surgery on a nontrivial knot $K$ yields the branched double cover of an alternating knot or link, then $|p/q|\leq 4g(K)+3$. This generalises a bound for lens space surgeries first established by Rasmussen. We also show…

We prove that 1) There exist infinitely many non-trivial codimension one "thick" knots in $\mathbb{R}^5$; 2) For each closed four-dimensional smooth manifold $M$ and for each sufficiently small positive $\epsilon$ the set of isometry…

A knot K in the 3-sphere is said to have Property nR if, whenever K is a component of an n-component link L and some integral surgery on L produces the connected sum of n copies of S^1 x S^2, there is a sequence of handle slides on L that…

We discuss an obstruction to a knot being smoothly slice that comes from minimum-genus bounds on smoothly embedded surfaces in definite 4-manifolds. As an example, we provide an alternate proof of the fact that the (2,1)-cable of the figure…

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;…

Given a diagram $D$ of a knot $K$, we consider the number $c(D)$ of crossings and the number $b(D)$ of overpasses of $D$. We show that, if $D$ is a diagram of a nontrivial knot $K$ whose number $c(D)$ of crossings is minimal, then…

Suppose $K$ is an unknot lying in the 1-skeleton of a triangulated 3-manifold with $t$ tetrahedra. Hass and Lagarias showed there is an upper bound, depending only on $t$, for the minimal number of elementary moves to untangle $K$. We give…

We consider surfaces embedded in a 3D contact sub-Riemannian manifold and the problem of the finiteness of the induced distance (i.e., the infimum of the length of horizontal curves that belong to the surface). Recently it has been proved…

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…

We consider manifold-knot pairs $(Y,K)$ where $Y$ is a homology sphere that bounds a homology ball. We show that the minimum genus of a PL surface $\Sigma$ in a homology ball $X$ such that $\partial (X, \Sigma) = (Y, K)$ can be arbitrarily…

Let K be a tame knot with irreducible exterior M(K) in a closed, connected, orientable 3--manifold Sigma such that pi_1(Sigma) is cyclic. If infinity is not a strict boundary slope, then the diameter of the set of strict boundary slopes of…

We prove that links with meridional rank 3 whose 2-fold branched covers are graph manifolds are 3-bridge links. This gives a partial answer to a question by S. Cappell and J. Shaneson on the relation between the bridge numbers and…

In this paper, we study on knots and closed incompressible surfaces in the 3-sphere via Morse functions. We show that both of knots and closed incompressible surfaces can be isotoped into a "related Morse position" simultaneously. As an…