arXiv research

Study cobordism distances between 3-braid links and trefoil knots.

Study revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links.

Study detects a specific type of link using annular Khovanov homology.

Paper proves homotopy braid group properties over integers and three strands.

We show that a certain linear representation of the singular braid monoid on three strands is faithful. Furthermore we will give a second - group theoretically motivated - solution to the word problem in this monoid.

A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c_3(…

New torsion patterns found in Khovanov homology of link diagrams.

Twisted links are obtained from a base link by starting with a $n$-braid representation, choosing several ($m$) adjacent strands, and applying one or more twists to the set. Various restrictions may be applied, e.g. the twists may be required to be positive or full twists, or the base braid may be required to have a ce…

We compute the Ozsváth-Szabó Heegaard Floer homology of three stranded pretzel knots.

Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function. We compute explicitly the geometric generating function for the group of braids o…

We exhibit a certain infinite family of three-stranded quasi-alternating pretzel knots which are counterexamples to Lobb's conjecture that the sl_3-knot concordance invariant s_3 (suitably normalised) should be equal to the Rasmussen invariant s_2. For this family, |s_3| < |s_2|. However, we also find other knots for w…

We show that the Artin pure braid group on at least four strands is not residually free. Our results also show that the pure braid group on at least three strands has corank two.

This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R=[2,1]. The project involves several steps: (i) parametrization of big families of knots a la arXiv:1506.00339, (ii) evaluating Racah/mixing matrices for various numbers of strands in var…

Kauffman and Lomonaco explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In the work of G. Alagic, M. Jarret, and S. Jordan it is shown that entanglement is a necessary condit…

The minimum number of colors is a challenging knot invariant since, by definition, its calculation requires taking the minimum over infinitely many minima. In this article we estimate and in some cases calculate the minimum number of colors for the Turk's head knots on three strands.

In this work we present a natural surjective map from rigid braids in B_3 (in Garside sense) to SL_2(N). This map provides an upper and a lower bound for the dilatation factor of a pseudo-Anosov 3-strand braid. These bounds only depend on the canonical length of the classical Garside structure of B_3.

Short note on braid index and quasipositivity of certain pretzel knots.

A knot is not squeezable if it fails to meet certain invariant criteria.

The study distinguishes knots using finite quotients of their fundamental groups.

In this article, we investigate various properties of the pure virtual braid group PV_3. From its canonical presentation, we obtain a free product decomposition of PV_3. As a consequence, we show that PV_3 is residually torsion free nilpotent, which implies that the set of finite type invariants in the sense of Goussar…

Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the crossing. In this paper we find a relationship between the triple crossing number and th…

The surface singular braid monoid corresponds to marked graph diagrams of knotted surfaces in braid form. In a quest to resolve linearity problem for this monoid, we will show that if it is defined on at least two or at least three strands, then its two or respectively three dimensional representations are not faithful…

We give a necessary, and in some cases sufficient, condition for sliceness inside the family of pretzel knots $P (p_1,...,p_n)$ with one $p_i$ even. The three stranded case yields two interesting families of examples: the first consists of knots for which the non-sliceness is detected by the Alexander polynomial while …

Explains Conway's tangle trick and its mathematical origins.

We study the Morton-Franks-Williams inequality for closures of simple braids (also known as positive permutation braids). This allows to prove, in a simple way, that the set of simple braids is a orthonormal basis for the inner product of the Hecke algebra of the braid group defined by Kálmán, who first obtained this r…

The paper classifies when certain graph braid groups are 3-manifold groups.

In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some $Ω_3$-moves that only differ in how the three strands that are involved in the move are ordered on the knot. To transform knot diagrams of isotopic knots into each other …

This paper is a new step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R^3->Q with all possible Q, for R=[3,1]. The calculation is made possible …

Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional topological quantum field theory representation of $\mathrm{SL}(2, \mathbb{Z})$ is a congruence subgroup. Motivated by their result, we explore when the kernel of an irreducible representation of the braid group $B_3$ with finite image enjoys a congruen…

Compactifies stability space for $A_2$ category, introducing $q$-deformed rational numbers.

Solves risk-sensitive investment via duality, entropic regularization, and RL.

Holomorphic maps between configuration spaces are classified, resolving quartic and elliptic curve problems.

The study shows that generative models can be effectively used to understand neural network performance.

The paper calculates actions of string link operations for 4- and 5-component links.

Study proves chainmail links are L-space links.

New findings on T-links derived from torus links.

Classifies colored links and spatial graphs up to colored link-homotopy.

Paper finds linking numbers for Montesinos links using a simple algorithm.

We define and prove properties of link lattice complexes for plumbed links.

Study of Lorenz links and T-links, showing equivalence and unique presentations.

Paper proves Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.

Extends positive and almost positive links to successively almost positive ones.

Study of knots and links in 2-complexes, defining linking numbers and polynomials.

A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams. It is further generalized to a twisted link. Twisted links are in one-to-one correspondence with stable equivalence classes of links in oriented thickenings of (possibly non-…

We generalized the periodic links to \emph{transitive} links in a $3$-manifold $M$. We find a complete classification theorem of transitive links in a $3$-dimensional sphere $\mathbb{R}^3$. We study these links from several different aspects including polynomial invariants using the relation between link polynomials of…

Characterizes a subset of links using quasipositive and homogeneous properties.

Study links' flat-virtual diagrams to create link invariants.

New link invariants from diagram colorings match link widths.