Related papers: A characterization of almost alternating knot
We show that among alternating knots, those which have diagrams whose Seifert and Tait graphs are isomorphic are dominant.
The study of a certain class of matrix integrals can be motivated by their interpretation as counting objects of knot theory such as alternating prime links, tangles or knots. The simplest such model is studied in detail and allows to…
We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded…
Knots are deeply entangled with every branch of science. One of the biggest open challenges in knot theory is to formalise a knot invariant that can unambiguously and efficiently distinguish any two knotted curves. Additionally, the…
A homogeneous knot is a generalization of alternating knots and positive knots. We determine the Rasmussen invariant of a homogeneous knot. This is a new class of knots such that the Rasmussen invariant is explicitly described in terms of…
The untwisting number of a knot K is the minimum number of null-homologous twists required to convert K to the unknot. Such a twist can be viewed as a generalization of a crossing change, since a classical crossing change can be effected by…
We complete the project begun by Callahan, Dean and Weeks to identify all knots whose complements are in the SnapPea census of hyperbolic manifolds with seven or fewer tetrahedra. Many of these ``simple'' hyperbolic knots have high crossing…
We give a simple characterization of braids that can be unplaited keeping separately their upper ends and their lower ends tied together
Weaving knots are alternating knots with the same projection as torus knots, and were conjectured by X.-S. Lin to be among the maximum volume knots for fixed crossing number. We provide the first asymptotically correct volume bounds for…
We derive a linear estimate of the signature of positive knots, in terms of their genus. As an application, we show that every knot concordance class contains at most finitely many positive knots.
We study the enumeration of alternating links and tangles, considered up to topological (flype) equivalences. A weight $n$ is given to each connected component, and in particular the limit $n\to 0$ yields information about (alternating)…
We show that a topological quantum computer based on the evaluation of a Witten-Reshetikhin-Turaev TQFT invariant of knots can always be arranged so that the knot diagrams with which one computes are diagrams of hyperbolic knots. The…
The knots-quivers correspondence states that various characteristics of a knot are encoded in the corresponding quiver and the moduli space of its representations. However, this correspondence is not a bijection: more than one quiver may be…
We construct two knot invariants. The first knot invariant is a sum constructed using linking numbers. The second is an invariant of flat knots and is a formal sum of flat knots obtained by smoothing pairs of crossings. This invariant can…
The large size limit of matrix integrals with quartic potential may be used to count alternating links and tangles. The removal of redundancies amounts to renormalizations of the potential. This extends into two directions: higher genus and…
The ropelength of a knot is the minimum length required to tie it. Computational upper bounds have previously been computed for every prime knot with up to 11 crossings. Here, we present ropelength measurements for the 2176 knots with 12…
In this article, we consider alternating knots on a closed surface in the 3-sphere, and show that these are not parallel to any closed surface disjoint from the prescribed one.
We use the G-signature theorem to define an invariant of strongly invertible knots analogous to the knot signature.
In this article the construction of a stationary random knot is proposed. The corresponding smooth random curve has no self-intersections in deterministic moments of time and changes its topological type at random moments.
Knot Floer homology is a knot invariant defined using holomorphic curves. In more recent work, taking cues from bordered Floer homology,the authors described another knot invariant, called "bordered knot Floer homology", which has an…