Related papers: A note on the concordance $\mathbb{Z}$-genus

The doubly slice genus of a knot in the 3-sphere is the minimal genus among unknotted orientable surfaces in the 4-sphere for which the knot arises as a cross-section. We use the classical signature function of the knot to give a new lower…

The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman's theorem that Alexander polynomial one knots are topologically slice. This paper develops…

Given a closed four-manifold $X$ with an indefinite intersection form, we consider smoothly embedded surfaces in $X \setminus $int$(B^4)$, with boundary a knot $K \subset S^3$. We give several methods to bound the genus of such surfaces in…

We discuss differences between genera of smooth and locally-flat non-orientable surfaces in the 4-ball with boundary a given torus knot or 2-bridge knot. In particular, we establish that a result by Batson on the smooth non-orientable…

This paper presents evidence supporting the surprising conjecture that in the topological category the slice genus of a satellite knot $P(K)$ is bounded above by the sum of the slice genera of $K$ and $P(U)$. Our main result establishes…

We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance…

J. Davis showed that the topological concordance class of a link in the 3-sphere is uniquely determined by its Alexander polynomial for 2-component links with Alexander polynomial one. A similar result for knots with Alexander polynomial…

Each ruling of a Legendrian link can be naturally treated as a surface. For knots, the ruling is 2-graded if and only if the surface is orientable. For 2-graded rulings of homogeneous (in particular, alternating) knots, we prove that the…

We show that the difference between the genus and the stable topological 4-genus of alternating knots is either zero or at least 1/3.

It is known that knot Floer homology detects the genus and Alexander polynomial of a knot. We investigate whether knot Floer homology of $K$ detects more structure of minimal genus Seifert surfaces for $K$. We define an invariant of…

We show that perturbing the definition of sl(n) Khovanov-Rozansky link homology gives a lower bound on the slice genus of a knot. As a corollary this yields another proof of Milnor's conjecture on the slice genus of torus knots.

In this article we define Lagrangian concordance of Legendrian knots, the analogue of smooth concordance of knots in the Legendrian category. In particular we study the relation of Lagrangian concordance under Legendrian isotopy. The focus…

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…

We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties.…

We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the…

We show that the difference between the Seifert genus and the topological 4-genus of a prime positive braid knot is bounded from below by an affine function of the minimal number of strands among positive braid representatives of the knot.…

Either fibered knots supporting the tight contact structure are unique in their smooth concordance class or there exists a fibered counterexample to the Slice-Ribbon Conjecture.

Let $M_K$ be the 2-fold branched cover of a knot $K in $S^3$. If $H_1(M_K) = {\bf Z}_3 \oplus {\bf Z}_{3^{2i}} \oplus G$ where 3 does not divide the order of $G$ then $K$ is not of order 4 in the concordance group. This obstruction detects…

Kronheimer and Mrowka asked whether the difference between the four-dimensional clasp number and the slice genus can be arbitrarily large. This question is answered affirmatively by studying a knot invariant derived from equivariant…

Let $X$ be a closed indefinite $4$-manifold with $b_+(X) = 3 \; ({\rm mod} \; 4)$ and with non-vanishing mod $2$ Seiberg--Witten invariants. We prove a new lower bound on the genus of a properly embedded surface in $X \setminus B^4$…