Related papers: Non-complex cobordisms between quasipositive knots

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 construct cobordisms of small genus between torus knots and use them to determine the cobordism distance between torus knots of small braid index. In fact, the cobordisms we construct arise as the intersection of a smooth algebraic curve…

Building off ideas developed by Agol, we construct a family of hyperbolic knots $K_n$ whose complements contain no closed incompressible surfaces and have Heegaard genus exactly $n$. These are the first known examples of small knots having…

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…

The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of…

We show there exist infinitely many knots of every fixed genus $g\geq 2$ which do not admit surgery to an L-space, despite resembling algebraic knots and L-space knots in general: they are algebraically concordant to the torus knot…

We consider the relationship between the crosscap number $\gamma$ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K \geq J$ if there exists an epimorphism…

We discuss relations among various positivities of knots and links, such as strong quasipositivity and quasipositivity. We give several pieces of supporting evidence for conjectural statements concerning these positivities and the defect of…

Ribbon concordances between knots generalize the notion of ribbon knots. Agol, building on work of Gordon, proved ribbon concordance gives a partial order on knots in $S^3$. In previous work, the author and Greene conjectured that positive…

For a connected cobordism S between two knots K1,K2 in S3, we establish an inequality involving the number of local maxima, the genus of S, and the torsion orders of Kht(K1),Kht(K2), where Kht denotes Lee's perturbation of Khovanov…

It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for…

The stable 4-genus of a knot K in 3-space is the limiting value of g_4(nK)/n, where g_4 denotes the 4-genus and n goes to infinity. This induces a seminorm on CQ, the concordance group tensored with the rational numbers. Basic properties of…

If a knot K in a closed, orientable 3-manifold M has a bridge surface T with distance at least 3 in the curve complex of T - K, then the genus of any essential surface in its exterior with non-empty, non-meridional boundary gives rise to an…

The unknotting number of a knot is bounded from below by its slice genus. It is a well-known fact that the genera and unknotting numbers of torus knots coincide. In this note we characterize quasipositive knots for which the genus bound is…

Let $T$ be a satellite knot, link, or spatial graph in a 3-manifold $M$ that is either $S^3$ or a lens space. Let $\mathfrak{b}_0$ and $\mathfrak{b}_1$ denote genus 0 and genus 1 bridge number, respectively. Suppose that $T$ has a companion…

We determine the locally flat cobordism distance between torus knots with small and large braid index, up to high precision. Here small means 2, 3, 4, or 6. As an application, we derive a surprising fact about torus knots that appear as…

An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen…

We show that if a link L with non-zero Alexander polynomial admits a locally flat cobordism to a `weakly m-split link', then the cobordism must have genus at least (m-1)/2. This generalises a recent result of J. Pardon.

In a lens space X of order r a knot K representing an element of the fundamental group pi_1 X = Z/rZ of order s <= r contains a connected orientable surface S properly embedded in its exterior X-N(K) such that the boundary of S intersects…

In this paper we study the cobordism of algebraic knots associated with weighted homogeneous polynomials, and in particular Brieskorn polynomials. Under some assumptions we prove that the associated algebraic knots are cobordant if and only…