Related papers: Singularity Knots of Minimal Surfaces in $\mathbb{…

We extend knot contact homology to a theory over the ring $\mathbb{Z}[\lambda^{\pm 1},\mu^{\pm 1}]$, with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in $S^3$ and can be…

Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from potentially singular complex algebraic surfaces and complex curves inside them. We prove that knot lattice…

An oriented compact 4-manifold $V$ with boundary $S^3$ is called a positon (resp. negaton) if its intersection form is positive definite (resp. negative definite) and it is simply connected. In this paper, we prove that there exist…

We show that if a classical knot diagram satisfies a certain combinatorial condition then it is minimal with respect to the number of classical crossings. This statement is proved by using the Kauffman bracket and the construction of atoms…

We prove that for any integer $n$ there exist infinitely many different knots in $S^3$ such that $n$-surgery on those knots yields the same 3-manifold. In particular, when $|n|=1$ homology spheres arise from these surgeries. This answers…

We study minimal surfaces in generic sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called {\it horizontal} area functional associated to the canonical…

With its boundary tracing out a link or knot in 3D, the Seifert surface is a 2D surface of core importance to topological classification. We propose the first-ever experimentally realistic setup where Seifert surfaces emerge as the boundary…

In (the surface of) a convex polytope P^3 in R^4, an area-minimizing surface avoids the vertices of P and crosses the edges orthogonally. In a smooth Riemannian manifold M with a group of isometries G, an area-minimizing G-invariant…

Suppose F is a compact orientable surface, K is a knot in F x I, and N is the 3-manifold obtained by some non-trivial surgery on K. If F x {0} compresses in N, then there is an annulus in F x I with one end K and the other end an essential…

Given a sequence of properly embedded minimal surfaces in a $3$-manifold with local bounds on area and genus, we prove subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with…

We study of the shape of a compact singular minimal surface in terms of the geometry of its boundary, asking what type of {\it a priori} information can be obtained on the surface from the knowledge of its boundary. We derive estimates of…

Under a simple assumption on Seifert surfaces, we characterise knots whose stable topological 4-genus coincides with the genus.

In this paper we are interested in symmetries of alternating knots, more precisely in those related to achirality. We call the following statement Tait's Conjecture on alternating -achiral knots: Let K be an alternating -achiral knot. Then…

For each integer m>1 and l>0 we construct a pair of compact embedded minimal surfaces of genus 1+4m(m-1)l. These surfaces desingularize the m Clifford tori meeting each other along a great circle at the angle of \pi/m. They are invariant…

Let C_T be the subgroup of the smooth knot concordance group generated by topologically slice knots and let C_D be the subgroup generated by knots with trivial Alexander polynomial. We prove the quotient C_T/C_D is infinitely generated, and…

Topological gauge theories in four dimensions which admit surface operators provide a natural framework for realizing homological knot invariants. Every such theory leads to an action of the braid group on branes on the corresponding moduli…

We present an algorithm that takes as input any element $B$ of the loop braid group and constructs a polynomial $f:\mathbb{R}^5\to\mathbb{R}^2$ such that the intersection of the vanishing set of $f$ and the unit 4-sphere contains the…

We use classical techniques to answer some questions raised by Daniele Celoria about almost-concordance of knots in arbitrary closed $3$-manifolds. We first prove that, given $Y^3 \neq S^3$, for any non-trivial element $g\in \pi_1(Y)$ there…

We give a simple obstruction for a knot to be amphichiral, in terms of the homology of the 2-fold branched cover. We work with unoriented knots, and so obstruct both positive and negative amphichirality.

We say that two knots are friends if they share the same 0-surgery. Two friends with different sliceness status would provide a counterexample to the 4-dimensional smooth Poincar\'e conjecture. Here we create a census of all friends with…