Related papers: Balanced algebraic unknotting, linking forms, and …

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…

For a prime knot group, the classifying space for the family of the subgroups generated by the meridians can be seen as an abstract analogue of the ambient manifold in which the knot lives. An explicit model of this ambient classifying…

We develop obstructions to a knot K in the 3-sphere bounding a smooth punctured Klein bottle in the 4-ball. The simplest of these is based on the linking form of the 2-fold branched cover of the 3-sphere branched over K. Stronger…

We introduce the notion of ascent sliceness of virtual knots. A representative of a virtual knot is an embedding $ S^1 \hookrightarrow \Sigma_{g} \times I $, for $ \Sigma_g $ a closed connected oriented surface of genus $ g $; the virtual…

Given a knot in $S^3$, one can associate to it a surface diffeomorphism in two different ways. First, an arbitrary knot in $S^{3}$ can be represented by braids, which can be thought of as diffeomorphisms of punctured disks. Second, if the…

We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary…

We consider a homology sphere $M_n(K_1,K_2)$ presented by two knots $K_1,K_2$ with linking number 1 and framing $(0,n)$. We call the manifold {\it Matsumoto's manifold}. We show that there exists no contractible bound of $M_n(T_{2,3},K_2)$…

We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, we deduce that if a 3-manifold Y admits a p^n-sheeted regular cover that is a Z/pZ-L-space…

A knot in the three-sphere is doubly slice if it is the cross-section of an unknotted two-sphere in the four-sphere. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We…

This paper provides a convenient and practical method to compute the homology and intersection pairing of a branched double cover of the 4-ball. To projections of links in the 3-ball, and to projections of surfaces in the 4-ball into the…

We establish a criterion that ensures a bounded almost complex curve in a bounded almost complex 4-manifold minimizes genus amongst all smooth surfaces that share its homology class and the transverse link on its boundary. An immediate…

A minimal knot is the intersection of a topologically embedded branched minimal disk in $\mathbb{R}^4$ $\mathbb{C}^2 $ with a small sphere centered at the branch point. When the lowest order terms in each coordinate component of the…

We construct ribbon surfaces of Euler characteristic one for several infinite families of alternating 3-braid closures. We also use a twisted Alexander polynomial obstruction to conclude the classification of smoothly slice knots which are…

For links with vanishing pairwise linking numbers, the link components bound pairwise disjoint surfaces in $B^{4}$. In this paper, we describe the set of genera of such surfaces in terms of the $h$-function, which is a link invariant from…

Given a (genus 2) cube-with-holes M, i.e. the complement in S^3 of a handlebody H, we relate intrinsic properties of M (like its cut number) with extrinsic features depending on the way the handlebody H is knotted in S^3. Starting from a…

Whitehead doubles provide a plethora of examples of knots that are topologically slice but not smoothly slice. We discuss the problem of the Whitehead double of the Figure 8 knot and survey commonly used techniques to obstructing sliceness.…

This note explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist…

In this article, we propose a new approach for describing and understanding knots and links in a 3-manifold through the use of an embedded non-orientable surface. Specifically, we define a plat-like representation based on this…

We desingularize a branch point $p$ of a minimal disk $F_0(\mathbb{D})$ in $\mathbb{R}^4$ through immersions $F_t$'s which have only transverse double points and are branched covers of the plane tangent to $F_0(\mathbb{D})$ at $p$. If $F_0$…

For n >1, if the Seifert form of a knotted 2n-1 sphere K in S^{2n+1} has a metabolizer, then the knot is slice. Casson and Gordon proved that this is false in dimension three (n = 1). However, in the three dimensional case it is true that…