arXiv research

New findings on T-links derived from torus links.

The paper classifies twisted torus links that are unlinks.

The study finds hyperbolic twisted torus links for certain twists.

New method calculates number of components in twisted torus links.

We introduce a new construction of a surface link in the 4-space. We construct a surface link as a branched covering over the standard torus, which we call a torus-covering link. We show that a certain torus-covering $T^2$-link is equivalent to the split union of spun $T^2$-links and turned spun $T^2$-links. We show th…

Study signatures of torus links and their cores using Neumann's equivariant signatures and Hirzebruch's formula.

Study the geometry of torus link character varieties, finding unexpected relations.

For an arbitrary positive integer $n$ and a pair $(p, q)$ of coprime integers, consider $n$ copies of a torus $(p,q)$ knot placed parallel to each other on the surface of the corresponding auxiliary torus: we call this assembly a torus $n$-link. We compute economical presentations of knot groups for torus links using t…

Study reveals $\mathfrak{sl}_2$ action on link homology of specific torus links.

We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun $T^2$-knots and turned spun $T^2$-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an u…

Classifies Legendrian torus and cable links, revealing symmetries and invariants.

We compute the triply graded Khovanov-Rozansky homology of a family of links, including positive torus links and $\operatorname{Sym}^l$-colored torus knots.

We introduce a new method for computing triply graded link homology, which is particularly well-adapted to torus links. Our main application is to the (n,n)-torus links, for which we give an exact answer for all n. In several cases, our computations verify conjectures of Gorsky et al relating homology of torus links wi…

Characters from logarithmic VOAs linked to torus link invariants.

We consider surface links in the 4-space which are presented by the form of simple branched coverings over the standard torus, which we call torus-covering links. In this paper, we study unknotting numbers of torus-covering links. In some cases, we can determine the unknotting numbers.

In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to …

Study fully augmented links in thickened torus, generalizing $S^3$ results.

Study Alexander polynomials of links in 3-torus.

Hyperbolic links in thickened torus decompose into angled tetrahedra.

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 twisted torus links and related generalizations. We determine upper bounds on their …

We compute Cayley graphs and automorphism groups for all finite $n$-quandles of two-bridge and torus knots and links, as well as torus links with an axis.

The present paper is an addendum to "Spherical structures on torus knots and links", arXiv:1008.0312, and concerns more general case of torus knot and link cone-manifolds.

New theorem for 4D links simplifies characterisation problem.

Researchers study chirality in a specific type of torus-covering link.

Researchers determine quantum filtration structure of torus links.

Jones polynomial for twisted torus knots is trivial if and only if the knot is trivial.

Classifies small links in an unmarked solid torus.

Study on quandle coloring quivers for (p, 2)-torus knots and links.

The paper extends knot polynomials to annular and toroidal pseudo links.

New link groups are derived from torus necklaces, connecting braid groups to reflection groups.

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

Alternative proof of Dynnikov's three-page index for torus links.

The Turaev genus and dealternating number of a link are two invariants that measure how far away a link is from alternating. We determine the Turaev genus of a torus knot with five or fewer strands either exactly or up to an error of at most one. We also determine the dealternating number of a torus knot with five or f…

We obtain the full list of Goeritz invariants of all torus knots and links.

Efficient cobordisms show minimal signature values on certain links.

We formulate the holographic principle for knots and links. For the "space" of all knots and links, torus knots T(2m+1,2) and torus links L(2m,2) play the role of the "boundary" of this space. Using the holographic principle, we find the skein relation of knots and links with the help of the recurrence relation for pol…

We prove a Torres-like formula for the $L^2$-Alexander torsions of links, as well as formulas for connected sums and cablings of links. Along the way we compute explicitly the $L^2$-Alexander torsions of torus links inside the three-sphere, the solid torus and the thickened torus.

Region crossing change for a knot or a proper link is an unknotting operation. In this paper, we provide a sharp upper bound on the region unknotting number for a large class of torus knots and proper links. Also, we discuss conditions on torus links to be proper.

The elliptic Hall algebra governs torus link homology.

The structure of the Khovanov homology of $(n,m)$ torus links has been extensively studied. In particular, Marko Stosic proved that the homology groups stabilize as $m\rightarrow\infty$. We show that the Khovanov homotopy types of $(n,m)$ torus links, as constructed by Robert Lipshitz and Sucharit Sarkar, also become s…

Study of panhandle polynomials of torus links with geometric applications.

Study reveals connection between torus links and logarithmic VOAs.

Symmetric function $L_{M,N}$ lifts torus link homology.

The study characterizes torus links' coloring quivers using dihedral quandles.

This article is about a natural distance function induced by smooth cobordisms between links. We show that the cobordism distance of torus links is determined by the profiles of their signature functions, up to a constant factor.

We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diag…

Triple-crossing number bound for knots and links, especially torus knots.

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