Related papers: Chebyshev Knots
We prove that 0 is a characterizing slope for infinitely many knots, namely the genus-1 knots whose knot Floer homology is 2-dimensional in the top Alexander grading, which we classified in recent work and which include all $(-3,3,2n+1)$…
We prove that the so-called t algebra of braids and ties supports a Markov trace. Further, by using this trace in the Jones' recipe, we define invariant polynomials for classical knots and singular knots. Our invariants have three…
The lattice stick number of a knot type is defined to be the minimal number of straight line segments required to construct a polygon presentation of the knot type in the cubic lattice. In this paper, we mathematically prove that the…
We show that for many classical knots one can find generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the…
The concordance group of algebraically slice knots is the subgroup of the classical knot concordance group formed by algebraically slice knots. Results of Casson and Gordon and of Jiang showed that this group contains in infinitely…
In this paper, we introduce a new nontrivial filtration, called F-order, for classical and virtual knot invariants; this filtration produces filtered knot invariants, which are called finite type invariants similar to Vassiliev knot…
A gordian unlink is a finite number of unknots that are not topologically linked, each with prescribed length and thickness, and that cannot be disentangled into the trivial link by an isotopy preserving length and thickness throughout. In…
We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are…
We exhibit infinitely many ribbon knots, each of which bounds infinitely many pairwise non-isotopic ribbon disks whose exteriors are diffeomorphic. This family provides a positive answer to a stronger version of an old question of Hitt and…
In this paper we use artificial neural networks to predict and help compute the values of certain knot invariants. In particular, we show that neural networks are able to predict when a knot is quasipositive with a high degree of accuracy.…
If a knot K has Seifert matrix V_K and has a prime power cyclic branched cover that is not a homology sphere, then there is an infinite family of non-concordant knots having Seifert matrix V_K.
A brief summary of the development of perturbative Chern-Simons gauge theory related to the theory of knots and links is presented. Emphasis is made on the progress achieved towards the determination of a general combinatorial expression…
This book is a detailed introduction to the theory of finite type (Vassiliev) knot invariants, with a stress on its combinatorial aspects. It is intended to serve both as a textbook for readers with no or little background in this area, and…
The A-polynomial of a knot is defined in terms of SL(2,C) representations of the knot group, and encodes information about essential surfaces in the knot complement. In 2005, Dunfield-Garoufalidis and Boyer-Zhang proved that it detects the…
A classical knot is described by a one-stroke trajectory with entanglements of a string. The replica method appears as a powerful tool in statistical mechanics for a polymer or self-avoiding walk. We consider this replica N to 0 limit in…
According to work of Hartley and Kawauchi in 1979 and 1980, the Conway Polynomial of all negative amphicheiral knots and strongly positive amphicheiral knots factors as $\phi(z)\phi(-z)$ for some $\phi(z)\in\mathbb Z[z]$. Moreover, a 2012…
The clasp number $c(K)$ of a knot $K$ is the minimum number of clasp singularities among all clasp disks bounded by $K$. It is known that the genus $g(K)$ and the unknotting number $u(K)$ are lower bounds of the clasp number, that is,…
In the present note, we will show that there are infinitely many composite twisted torus knots.
There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every…
It is known that every nontrivial knot has at least two quadrisecants. Given a knot, we mark each intersection point of each of its quadrisecants. Replacing each subarc between two nearby marked points with a straight line segment joining…