Related papers: Small knots of large Heegaard genus

Using a combinatorial approach described in a recent paper of Manolescu, Ozsv\'ath, and Sarkar we compute the Heegaard-Floer knot homology of all knots with at most 12 crossings as well as the $\tau$ invariant for knots through 11…

We consider the family of Heegaard splittings of genus $g$ at least three which are defined via a glueing map that is the $n$-th power of the Dehn twist along a curve that satisfies a natural topological assumption, namely pared…

It was conjectured by Lopez that every closed irreducible non-Haken 3-manifold contains a small knot. In this paper, we give explicit examples of hyperbolic small knots in most closed orientable spherical 3-manifolds other than prism…

A well-known conjecture in knot theory says that the percentage of hyperbolic knots amongst all of the prime knots of $n$ or fewer crossings approaches $100$ as $n$ approaches infinity. In this paper, it is proved that this conjecture…

We use the methods of Hedden, Juhasz, and Sarkar to exhibit a set of arborescent knots that bound large numbers of non-isotopic minimal genus spanning surfaces. In particular, we describe a sequence of prime knots K_{n} which will bound at…

If K is a rationally null-homologous knot in a 3-manifold M, the rational genus of K is the infimum of -\chi(S)/2p over all embedded orientable surfaces S in the complement of K whose boundary wraps p times around K for some p (hereafter: S…

Let $h(K)$, $g_H(K)$, $g_1(K)$, $t(K)$ be the $h$-genus, Heegaard genus, bridge-1 genus, tunnel number of a knot $K$ in the $3$-sphere $S^3$, respectively. It is known that $g_H(K)-1=t(K)\leq g_1(K)\leq h(K)\leq g_H(K)$. A natural question…

We use Dehn surgery methods to construct infinite families of hyperbolic knots in the 3-sphere satisfying a weak form of the Turaev--Viro invariants volume conjecture. The results have applications to a conjecture of Andersen, Masbaum, and…

Let $K_1, K_2$ be two knots with $t(K_1)+t(K_2)>2$ and $t(K_1 # K_2)=2$. Then, in the present paper, we will show that any genus three Heegaard splittings of $E(K_1 # K_2)$ is strongly irreducible and that $E(K_1 # K_2)$ has at most four…

We complete the project begun by Callahan, Dean and Weeks to identify all knots whose complements are in the SnapPea census of hyperbolic manifolds with seven or fewer tetrahedra. Many of these ``simple'' hyperbolic knots have high crossing…

The concordance genus of a knot K is the minimum three-genus among all knots concordant to K. For prime knots of 10 or fewer crossings there have been three knots for which the concordance genus was unknown. Those three cases are now…

We answer a question posed by Fielder in [1] concerning two notions of crossing number for algebraic knots $K$ under Hopf fibration, one topological, denoted $h(K)$, the other coming from the realization of such knots around complex…

In knot Floer homology, there are two types of torsion order. One is the minimal power of the action of the variable $U$ to annihilate the $\mathbb{F}_2[U]$-torsion submodule of the minus version of knot Floer homology $\mathrm{HFK}^-(K)$.…

We bound the hyperbolic volumes of a large class of knots and links, called homogeneously adequate knots and links, in terms of their diagrams. To do so, we use the decomposition of these links into ideal polyhedra, developed by Futer,…

We construct a hyperbolic 3-manifold $M$ (with $\partial M$ totally geodesic) which contains no essential closed surfaces, but for any even integer $g> 0$ there are infinitely many separating slopes $r$ on $\partial M$ so that $M[r]$, the…

We identify all hyperbolic knots whose complements are in the census of orientable one-cusped hyperbolic manifolds with eight ideal tetrahedra. We also compute their Jones polynomials.

Suppose $K$ is a knot in $S^3$ with bridge number $n$ and bridge distance greater than $2n$. We show that there are at most ${2n\choose n}$ distinct minimal genus Heegaard splittings of $S^3\setminus\eta(K)$. These splittings can be divided…

We consider closed acylindrical surfaces in 3-manifolds and in knot and link complements, and show that the genus of these surfaces is bounded linearly by the number of tetrahedra in the triangulation of the manifold and by the number of…

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) <=…

The slicing number of a knot, $u_s(K)$, is the minimum number of crossing changes required to convert $K$ to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus $g_s(K)$. We show that for many…