Related papers: Links and Hurwitz curves

It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower…

In an earlier note [arXiv:2301.00295] it was shown that there is an upper bound to the number of disjoint Hopf links (and certain related links) that can be embedded in the unit cube where there is a fixed separation required between the…

Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. We study the geometry of…

We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number…

We establish an upper bound for the Thurston-Bennequin number of a Legendrian link using the Khovanov homology of the underlying topological link. This bound is sharp in particular for all alternating links, and knots with nine or fewer…

A differential geometric characterization of the braid-index of a link is found. After multiplication by 2pi, it equals the infimum of the sum of total curvature and total absolute torsion over holonomic representatives of the link. Upper…

In a recent paper, McMullen showed an inequality between the Thurston norm and the Alexander norm of a 3-manifold. This generalizes the well-known fact that twice the genus of a knot is bounded from below by the degree of the Alexander…

A special class of braids, called woven, is introduced and it is shown that every conjugation class of the braid group contains woven braids. In consequence, links can be presented as plats or closures of woven braids. Restricting on knots,…

Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between…

In classical knot theory, Markov's theorem gives a way of describing all braids with isotopic closures as links in $\mathbb{R}^3$. We present a version of Markov's theorem for extended loop braids with closure in $B^3 \times S^1$, as a…

The slope conjecture gives a precise relation between the degree of the colored Jones polynomial of a knot and the boundary slopes of essential surfaces in the knot complement. In this note we propose a generalization of the slope…

It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known…

In this paper, we find a more straightforward problem that is equivalent to one of the major challenges in knot theory: the classification of links in the 3-sphere. More precisely, we provide a simpler braid description for all links in the…

We prove that any diagram of the unknot with c crossings may be reduced to the trivial diagram using at most (236 c)^{11} Reidemeister moves. Moreover, every diagram in this sequence has at most (7 c)^2 crossings. We also prove a similar…

Extended welded links are a generalization of Fenn, Rim\'{a}nyi, and Rourke's welded links. Their braided counterpart are extended welded braids, which are closely related to ribbon braids and loop braids. In this paper we prove versions of…

Twisted knot theory, introduced by M.O. Bourgoin, is a generalization of virtual knot theory. It naturally yields the notion of a twisted braid, which is closely related to the notion of a virtual braid due to Kauffman. In this paper, we…

We give a simple unified proof for several disparate bounds on Thurston-Bennequin number for Legendrian knots and self-linking number for transverse knots in R^3, and provide a template for possible future bounds. As an application, we give…

We prove that a nicely fibered link (by which we mean the binding of an open book) in a tight contact manifold $(M,\xi)$ with zero Giroux torsion has a transverse representative realizing the Bennequin bound if and only if the contact…

For a positive braid link, a link represented as a closed positive braids, we determine the first few coefficients of its HOMFLY polynomial in terms of geometric invariants such as, the maximum euler characteristics, the number of split…

Given a thin strip of paper, tie a knot, connect the ends, and flatten into the plane. This is a physical model of a folded ribbon knot in the plane, first introduced by Louis Kauffman. We study the folded ribbonlength of these folded…