Related papers: The Bennequin number of n-trivial closed n-braids …
The affine Hecke algebra $\dot H_n$ of type $A$ is often presented as a quotient of the braid algebra of $n$-braids in the annulus. This leads to diagrammatic representations in terms of braids in the annulus, subject to a quadratic…
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 this note, we study non-standard models of the rational numbers with countably many elements. These are ordered fields, and so it makes sense to complete them, using non-standard Cauchy sequences. The main result of this note shows that…
Starting from the free field realization of Kac-Moody Lie algebra, we define a generalized Yang-Yang function. Then for the Lie algebra of type $A_{n}$, we derive braiding and fusion matrix by braiding the thimble from the generalized…
We give a complete classification of homomorphisms from the braid group on $n$ strands to the braid group on $2n$ strands when $n$ is at least 5. We also classify endomorphisms of the braid group on 4 strands, as well as homomorphisms from…
We provide linear lower bounds for the signature of positive braids in terms of the three genus of their braid closure. This yields linear bounds for the topological slice genus of knots that arise as closures of positive braids.
Using the band representation of the 3-strand braid group, it is shown that the genus of 3-braid links can be read off their skein polynomial. Some applications are given, in particular a simple proof of Morton's conjectured inequality and…
We use some Lie group theory and Budney's unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the…
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 show that the totally nonnegative part of the twisted product of flag varieties of a Kac-Moody group admits a cellular decomposition, and the closure of each cell is a topological manifold with boundary. We also establish explicit…
Minimum braids are a complete invariant of knots and links. This paper defines minimum braids, describes how they can be generated, presents tables for knots up to ten crossings and oriented links up to nine crossings, and uses minimum…
A transverse knot is a knot that is transverse to the planes of the standard contact structure on real 3-space. In this paper we prove the Markov Theorem for transverse braids, which states that two transverse closed braids that are…
The crossing matrix of a braid on $N$ strands is the $N\times N$ integer matrix with zero diagonal whose $i,j$ entry is the algebraic number (positive minus negative) of crossings by strand $i$ over strand $j$ . When restricted to the…
The genus non-increasing totally positive unknotting number is the minimum number of crossing changes that transform a knot into the unknot, such that all the crossing changes are positive-to-negative crossing changes that do not increase…
The third unramified cohomology group is shown to vanish on certain varieties equipped with a pencil of quadrics or of smooth complete intersections of two quadrics. Over the complex field, this shows that the integral Hodge conjecture in…
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,…
This paper presents a nonnegative polynomial that cannot be represented with nonnegative coefficients in the simplicial Bernstein basis by subdividing the standard simplex. The example shows that Bernstein Theorem cannot be extended to…
We prove that under fairly general conditions an iterated exchange move gives infinitely many non-conjugate braids. As a consequence, every knot has infinitely many conjugacy classes of n-braid representations if and only if it has one…
We showed that the order of torsion homology classes in the grid homology of a knot is a lower bound for the unknotting number.
We introduce and characterise grid classes, which are natural generalisations of other well-studied permutation classes. This characterisation allows us to give a new, short proof of the Fibonacci dichotomy: the number of permutations of…