Related papers: Knots with small rational genus

In this paper, we define the primitive/Seifert-fibered property for a knot in S^3. If satisfied, the property ensures that the knot has a Dehn surgery that yields a small Seifert-fibered space (i.e. base S^2 and three or fewer critical…

We define a knot to be half ribbon if it is the cross-section of a ribbon 2-knot, and observe that ribbon implies half ribbon implies slice. We introduce the half ribbon genus of a knot K, the minimum genus of a ribbon knotted surface of…

Given a normal surface singularity (X,0), its link, M is a closed differentiable three dimensional manifold which carries much analytic information. It is an interesting question to ask whether, under suitable analytic and topological…

In knot concordance three genera arise naturally, g(K), g_4(K), and g_c(K): these are the classical genus, the 4-ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to K. Clearly 0 <= g_4(K) <=…

In the early 2000s, Frigerio, Martelli, and Petronio studied $3$-manifolds of smallest combinatorial complexity that admit hyperbolic structures. As part of this work they defined and studied the class $M_{g,k}$ of smallest complexity…

Given a knot $K$ in a closed orientable manifold $M$ we define the growth rate of the tunnel number of $K$ to be $gr_t(K) = \limsup_{n \to \infty} \frac{t(nK) - n t(K)}{n-1}$. As our main result we prove that the Heegaard genus of $M$ is…

Let $p: S\to S_g$ be a finite covering of an orientable closed surface of genus $g$. We prove that, for $g\geq 3$, the rational homology group $H_1(S;{\mathbb Q})$ is generated by cycles supported on simple closed curves $\gamma\subset S$…

For $d\geq 2$, the regular genus of a closed connected PL $d$-manifold $M$ is the least genus (resp., half of the genus) of an orientable (resp., a non-orientable) surface into which a crystallization of $M$ imbeds regularly. The regular…

A Heegaard diagram for a 3-manifold M is a closed, oriented surface S together with a pair (X, Y) of compact 1-manifolds in S whose components serve as attaching curves for the 2-handles of the two sides of a Heegaard splitting for M. The…

A symplectic manifold is called symplectic rationally connected if there is a non-zero genus zero Gromov-Witten invariant with two point insertions. It is conjectured that every smooth projective rationally connected variety is symplectic…

We prove that various families of energies relevant in geometric knot theory satisfy the Palais-Smale condition (PS) on submanifolds of arclength para\-metrized knots. These energies include linear combinations of the Euler-Bernoulli…

We prove that a simple knot in the lens space $L(p,q)$ fibers if and only if its order in homology does not divide any remainder occurring in the Euclidean algorithm applied to the pair $(p,q)$. One corollary is that if $p=m^2$ is a perfect…

Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual…

This paper gives a partial description of the homotopy type of K, the space of long knots in 3-dimensional Euclidean space. The primary result is the construction of a homotopy equivalence between K and the free little 2-cubes object over…

We describe four hyperbolic knot complements in $\mathbb{S}^3$, each of which covers a prism orbifold: the quotient of $\mathbb{H}^3$ by the action of a discrete group generated by reflections in the faces of a polyhedron that has the…

We show that the Vassiliev invariants of orders $\leq n$ of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its…

We show that the equivariant and non-equivariant non-orientable 4-genus of p-periodic knots may differ, for any choice of p>1. Similar results have previously been obtained for the smooth 4-genus and non-orientable 3-genus of a periodic…

A knot $K$ is called $(m,n)$-fertile if for every prime knot $K'$ whose crossing number is less than or equal to $m$, there exists an $n$-crossing diagram of $K$ such that one can get $K'$ from the diagram by changing its over-under…

The $k$th module of a surface-knot of a genus $g$ in the 4-sphere is the $k$th integral homology module of the infinite cyclic covering of the surface-knot complement. The reduced first module is the quotient module of the first module by…

Let $LHT$ be a left handed trefoil knot and $K$ be any knot. We define $M_n(K)$ to be the homology $3$-sphere which is represented by a simple link of $LHT$ and $LHT \sharp K$ with framings $0$ and $n$ respectively. Starting with this link,…