Related papers: Chebotarev links are stably generic

Let $G$ be a graph with $n$ vertices, $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$ and $I(G)$ denote the edge ideal of $G$. For every collection $\mathcal{H}$ of connected graphs with…

When $G$ is abelian and $l$ is a prime we show how elements of the relative K-group $K_{0}({\bf Z}_{l}[G], {\bf Q}_{l})$ give rise to annihilator/Fitting ideal relations of certain associated ${\bf Z}[G]$-modules. Examples of this…

We establish a novel connection between algebraic number theory and knot theory. We show that the number of equivalence classes of integral binary quadratic forms of discriminant $t^2 - 4$ (for $t\neq \pm 2$) is equal to the number of…

Let $\phi : S^1\times D^2\to S^1$ be the natural projection. An oriented knot $K\hookrightarrow V = S^1\times D^2$ is called an almost closed braid if the restriction of $\phi$ to K has exactly two (non-degenerate) critical points (and K is…

We prove that if $G$ is the graph of a connected triangulated $(d-1)$-manifold, for $d\geq 3$, then $G$ is generically globally rigid in $\mathbb R^d$ if and only if it is $(d+1)$-connected and, if $d=3$, $G$ is not planar. The special case…

For a prime number $q\neq 2$ and $r>0$ we study, whether there exists an isometry of order $q^r$ acting on a free $\mathbb{Z}_{p^k}$-module equipped with a scalar product. We investigate, whether there exists such an isometry with no…

It has been known that any Alexander polynomial of a knot can be realized by a quasipositive knot. As a consequence, the Alexander polynomial cannot detect quasipositivity. In this paper we prove a similar result about Vassiliev invariants:…

We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include new proofs of Krcatovich's result that knots with $L$-space surgeries are prime and Hedden and…

Let $A$ be the ring of integers of a number field $K$. Let $G \subseteq GL_3(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y, Z]$ (fixing $A$) and let $S = R^G$ be the ring of invariants. Assume the Veronese subring $S^{<m>}$ of…

Given a real analytic function $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated critical point at the origin, the link $L_f$ of the singularity is a real fibred knot in $\mathbb{S}^{3}$. From this singularities, we construct a family…

In several different settings, one comes across situations in which the objects of study are locally consistent but globally inconsistent. Earlier work about probability distributions by Vorob'ev (1962) and about database relations by…

Geometric aspects of the filtration on classical links by k-quasi-isotopy are discussed, including the effect of Whitehead doubling, relations with Smythe's n-splitting and Kobayashi's k-contractibility. One observation is:…

Grid diagrams encode useful geometric information about knots in S^3. In particular, they can be used to combinatorially define the knot Floer homology of a knot K in S^3, and they have a straightforward connection to Legendrian…

We introduce a special class of knots, called global knots, in F^2 x R and we construct new isotopy invariants, called T-invariants, for global knots. Some T-invariants are of finite type but they cannot be extracted from the generalized…

We find further evidence for the conjecture relating large N Chern-Simons theory on S^3 with topological string on the resolved conifold geometry by showing that the Wilson loop observable of a simple knot on S^3 (for any representation)…

A knot in $S^3$ is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all…

We establish a connection between knot theory and cluster algebras via representation theory. To every knot diagram (or link diagram), we associate a cluster algebra by constructing a quiver with potential. The rank of the cluster algebra…

For every link $L$ we construct a complex algebraic plane curve that intersects $S^3$ transversally in a link $\tilde{L}$ that contains $L$ as a sublink. This construction proves that every link $L$ is the sublink of a quasipositive link…

The relative Novikov conjecture states that the relative higher signatures of manifolds with boundary are invariant under orientation-preserving homotopy equivalences of pairs. In this paper, we study the relative Baum-Connes assembly map…

Let $\Sigma_0,\Sigma_1$ be closed oriented surfaces. Two oriented knots $K_0 \subset \Sigma_0 \times [0,1]$ and $K_1 \subset \Sigma_1 \times [0,1]$ are said to be (virtually) concordant if there is a compact oriented $3$-manifold $W$ and a…