arXiv research

Algorithm simplifies Khovanov homology computations for 4-strand torus links.

The family of negative torus links $T_{p,q}$ over a fixed number of strands $p$ admits a stable limit in reduced Khovanov homology as $q$ grows to infinity. In this paper, we endow this stable space with a bi-graded commutative algebra structure. We describe these algebras explicitly for $p=2,3,4$. As an application, w…

Computes a specific homology for a type of braid.

We prove that the signature bound for the topological 4-genus of 3-strand torus knots is sharp, using McCoy's twisting method. We also show that the bound is off by at most 1 for 4-strand and 6-strand torus knots, and improve the upper bound on the asymptotic ratio between the topological 4-genus and the Seifert genus …

Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's diagonalization theorem. As a consequence, we prove the slice-ribbon conjecture for 4-strand…

Circle graph complexes reveal link properties via Khovanov homology.

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 unitary group. This implies that, except possibly for closures of full-twist braids, …

Study shows torsion in knot module for specific Montesinos knots.

Researchers derived Kauffman bracket polynomial for Celtic link shadows using two methods.

The study classifies $χ-$slice pretzel links and Seifert fiber spaces.

New findings on T-links derived from torus links.

The paper classifies twisted torus links that are unlinks.

Using an obstruction based on Donaldson's theorem on the intersection forms of definite 4-manifolds, we determine which connected sums of lens spaces smoothly embed in S^4. We also find constraints on the Seifert invariants of Seifert 3-manifolds which embed in S^4 when either the base orbifold is non-orientable or the…

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.