arXiv research

Homogeneous quasipositive links are positive if their Seifert circles match braid index.

We show that there are only finitely many homogeneous links whose Conway polynomial has any given degree. Using this we give an example of an inhomogeneous, fibred knot. Secondly, we show how to compute the monodromy of a homogeneous link complement from a homogeneous braid word representative.

New bounds on HOMFLY polynomial for homogeneous links.

Characterizes a subset of links using quasipositive and homogeneous properties.

New minimal link diagrams found, including torus links and homogeneous ones.

In this paper we review the definitions of homogeneous and alternative links. We also give two new characterizations of an alternative link diagram, one within the context of the enhanced checkerboard graph and another from the labeled Seifert graph.

The twisting technique creates infinite links.

We characterize positive links in terms of strong quasipositivity, homogeneity and the value of Rasmussen, Beliakova and Wehrli's $s$-invariant. We also study almost positive links, in particular, determine the $s$-invariants of almost positive links. This result suggests that all almost positive links might be strongl…

The paper classifies virtual links using the arc shift operation.

Proof confirms conjecture for certain braids and their closures.

We bound the hyperbolic volumes of a large class of knots and links, called homogeneously adequate knots and links, in terms of their diagrams. To do so, we use the decomposition of these links into ideal polyhedra, developed by Futer, Kalfagianni, and Purcell. We identify essential product disks in these polyhedra.

Homogeneous links were introduced by Peter Cromwell, who proved that the projection surface of these links, that given by the Seifert algorithm, has minimal genus. Here we provide a different proof, with a geometric rather than combinatorial flavor. To do this, we first show a direct relation between the Seifert matrix…

Is any positive knot the closure of a positive braid? No. But if we consider positivity in terms of the generators of the braid group due to Birman, Ko and Lee, then the answer is yes. In this paper we prove that the same occurs when considering homogeneity. In the way we prove that the plumbing of two surfaces is a BK…

We give a geometric proof of the following result of Juhasz. \emph{Let $a_g$ be the leading coefficient of the Alexander polynomial of an alternating knot $K$. If $|a_g|<4$ then $K$ has a unique minimal genus Seifert surface.} In doing so, we are able to generalise the result, replacing `minimal genus' with `incompress…

In this paper, we characterize the sigma-adequacy of a link diagram in two ways: in terms of a certain edge subset of its Tait graph and in terms of a certain product of Tutte polynomials. Furthermore, we show that the symmetrized Tutte polynomial of the Tait graph of a link diagram can be written as a sum of these pro…

Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.

We are going to use the Euler's vector fields in order to show that for real quasi-homogeneous singularities with isolated critical value, the Milnor's fibration in a "thin" hollowed tube involving the zero level and the fibration in the complement of "link" in sphere are equivalents, since they exist. Moreover, in ord…

Innovative rack theory applied to Legendrian links.

In the present paper, we study deformations of polar weighted homogeneous polynomials which are also polar weighted homogeneous polynomials. We describe a round handle decomposition of the Milnor fibration of a deformation of a polar weighted homogeneous polynomial concretely and give the number of round handles by the…

Flat semigroups can represent normal weighted homogeneous surface singularities.

We compute the $p$-primary components of the linking pairings of orientable 3-manifolds admitting a fixed-point free $S^1$-action. Using this, we show that any non-singular linking pairing on a finite abelian group with homogeneous 2-primary summand is realized by such a manifold. However, some pairings on inhomogeneou…

We generalize the Khovanov-Rozansky cohomology for n=2 by means of a homogeneous potential that depends on two parameters, to obtain the universal Khovanov-Rozansky sl(2) link cohomology. This theory is equivalent to the universal foam sl(2) link cohomology, after tensoring both theories with appropriate rings.

New solutions found for a complex boundary problem.

We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…

A new method detects communities in multi-relational networks.

This paper continues our study, initiated in [arXiv:1108.3370], of essential state surfaces in link complements that satisfy a mild diagrammatic hypothesis (homogeneously adequate). For hyperbolic links, we show that the geometric type of these surfaces in the Thurston trichotomy is completely determined by a simple gr…

We study a canonical spanning surface obtained from a knot or link diagram depending on a given Kauffman state, and give a sufficient condition for the surface to be essential. By using the essential surface, we can see the triviality and splittability of a knot or link from its diagrams. This has been done on the exte…

Ricci curvature links volume convexity and minimal submanifolds.

Unique steady and expanding solitons with spherical links identified.

We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds wit…

New Stein fillings found for non-weighted homogeneous singularities.

In the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds. The main feature of our approach is to describe the Cartan-Ehresmann connection (gauge field) for principal circle bundles over complex flag manifolds by using elements of representation theory …

Every Kauffman state σof a link diagram D(K) naturally defines a state surface S_σwhose boundary is K. For a homogeneous state σ, we show that K is a fibered link with fiber surface S_σif and only if an associated graph G'_σis a tree. As a corollary, it follows that for an adequate knot or link, the second and next-to-…

In this paper we define alternating Kauffman states of links and we characterize when the induced state surface is a fiber. In addition, we give a different proof of a similar theorem of Futer, Kalfagianni and Purcell on homogeneous states.

Linking numbers of modular knots derived from geometric and algebraic properties.

Study the relationship between braids formed by roots and critical points of polynomials.

We prove that the quasi-homogenous symbols on the projective space $\mathbb{P}^n(\mathbb{C})$ yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball $\mathbb{B}^n$. These algebras are Banach but not $C^*$. We prove the existen…

We analyse second order (in Riemann curvature) geometric flows (un-normalised) on locally homogeneous three manifolds and look for specific features through the solutions (analytic whereever possible, otherwise numerical) of the evolution equations. Several novelties appear in the context of scale factor evolution, fix…

An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, $4$-genus and $3$-genus of a positive knot are equal. In this paper, we p…

In a spatially embedded network, that is a network where nodes can be uniquely determined in a system of coordinates, links' weights might be affected by metric distances coupling every pair of nodes (dyads). In order to assess to what extent metric distances affect relationships (link's weights) in a spatially embedde…

We show that if a braid $B$ can be parametrised in a certain way, then previous work can be extended to a construction of a polynomial $f:\mathbb{R}^4\to\mathbb{R}^2$ with the closure of $B$ as the link of an isolated singularity of $f$, showing that the closure of $B$ is real algebraic. In particular, we prove that cl…

Let P be a closed smooth (4j-2)-connected 8j-manifold. We complete Wilkens' classification of the manifolds P for j = 1,2 and give an alternative proof to Wall's classification of the manifolds for j > 2. The Hopf-invariant-one dimensions (j=1,2) are characteristed by the fact that the quadratic linking functions which…

Let $L_f$ be a link of an isolated hypersurface singularity defined by a weighted homogenous polynomial $f.$ In this article, we give ten examples of $2$-connected seven dimensional Sasaki-Einstein manifolds $L_f$ for which $H_{3}(L_f, \mathbb{Z})$ is completely determined. Using the Boyer-Galicki construction of links…

Bayesian MS-VAR model for pricing equity-linked life insurance products.

Study on homology groups of cDV singularity links, identifying their topology.

Torsion and Betti numbers for knots are special cases of more general invariants associated to a finitely generated group G and epimorphism from G to the integers. The sequence of Betti numbers is always periodic; under mild hypotheses, the sequence of torsion numbers satisfies a linear homogeneous recurrence relation …

It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "s…

We show that for a special alternating link diagram, the following three polynomials are essentially the same: a) the part of the HOMFLY polynomial that corresponds to the leading term in the Alexander polynomial; b) the $h$-vector for a triangulation of the root polytope of the Seifert graph and c) the enumerator of p…