arXiv research

We show that if a split link is obtained from a split link $L$ in $S^3$ by $1/n$-Dehn surgery along a trivial knot $C$, then the link $L\cup C$ is splittable. That is to say, it is impossible to obtain a split link from a split link via a non-trivial twisting. As its corollary, we completely determine when a trivial li…

New methods convert complex link presentations to simpler, recognizable forms.

Link Floer homology detects split links.

The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the sp…

New spheres can split a 4D link in ways not possible in 3D.

Prime homology detects split links in prime characteristic.

New methods for delta-moves on algebraically split links identified.

The splitting number of a link is the minimum number of crossing changes between distinct components that is required to convert the link into a split link. We provide a bound on the splitting number in terms of the four-genus of related knots.

In this paper, we define a lassoing on a link, a local addition of a trivial knot to a link. Let K be an s-component link with the Conway polynomial non-zero. Let L be a link which is obtained from K by r-iterated lassoings. The complete splitting number split(L) is greater than or equal to r+s-1, and less than or equa…

The paper extends keenness concept to bridge splittings and finds conditions for existence.

Same-genus Seifert surfaces for non-split alternating links are smoothly isotopic.

New links split by integer homology spheres but not by others.

We study the Kakimizu complex of a split link. As part of this, we also study Seifert surfaces and the Kakimizu complex for a non-split link in a 3-ball. In addition, we show that a simplex of the Kakimizu complex of a non-split link can be realised in an essentially unique way.

The paper studies splitting maps in link Floer homology using skein exact sequences.

Study shows how Khovanov homology behaves for split links and cobordisms.

The paper examines when 2-string tangles can be embedded into specific link types.

Study shows that splitting links requires an arbitrarily large number of extra crossings.

Physical knots and links are one-dimensional submanifolds of R^3 with fixed length and thickness. We show that isotopy classes in this category can differ from those of classical knot and link theory. In particular we exhibit a Gordian Split Link, a two component link that is split in the classical theory but cannot be…

Delta-unlinking number measures how to unlink algebraically split links.

Boring is an operation which converts a knot or two-component link in a 3--manifold into another knot or two-component link. It generalizes rational tangle replacement and can be described as a type of 2--handle attachment. Sutured manifold theory is used to study the existence of essential spheres and planar surfaces …

The splitting number of a link is the minimal number of crossing changes between different components required to convert it into a split link. We obtain a lower bound on the splitting number in terms of the (multivariable) signature and nullity. Although very elementary and easy to compute, this bound turns out to be …

New results on splitting tangles and spatial graphs.

New examples show limits of physical link isotopies.

We investigate the concordance properties of `parallel links' P(K), given by the (2,0) cable of a knot K. We focus on the question: if P(K) is concordant to a split link, is K necessarily slice? We show that if P(K) is smoothly concordant to a split link, then many smooth concordance invariants of K must vanish, includ…

We construct an infinite family of topologically slice 2--component boundary links $\ell_i$, none of which is smoothly concordant to a split link, such that $g_4(\ell_i)=i$.

Construct non-split 2-component links in $S^4$ to produce exotic pairs of 4-manifolds

Extends classical results to virtual links, proving new properties of alternating and semi-alternating virtual links.

New bounds on Heegaard distance for link complements in 3D space.

We provide necessary conditions for the Alexander polynomials of algebraically split component-preservingly amphicheiral links. We raise a conjecture that the Alexander polynomial of an algebraically split component-preservingly amphicheiral link with even components is zero. Our necessary conditions and some examples …

When a Dehn filled link manifold contains a geometrically incompressible one-sided surface, it is shown there is a unique boundary incompressible position that the surface can take in the link space. The proof uses a version of the sweep-out technique from two-sided Heegaard splitting theory. When applied to one-sided …

New deformation of link homology for colored diagrams.

We provide an algorithm to determine whether a link L admits a crossing change that turns it into a split link, under some fairly mild hypotheses on L. The algorithm also provides a complete list of all such crossing changes. It can therefore also determine whether the unlinking number of L is 1.

We define a family of link concordance invariants $\left\{ s_n \right\}_{n=2,3, \cdots}$. These link concordance invariants give lower bounds on the slice genus of a link $L$. We compute the slice genus of positive links. Moreover, these invariants give lower bounds on the link splitting number of a link. Especially, t…

We show that if $K$ is a nontrivial knot then the proportion of satellites of $K$ among all of the prime non-split links of $n$ or fewer crossings does not converge to $0$ as $n$ approaches infinity. This implies in particular that the proportion of hyperbolic links among all of the prime non-split links of $n$ or fewe…

Study the space of embeddings of split links in 3D and 4D.

Researchers create infinite Brunnian links of 3-balls in 4-sphere.

The weak splitting number $wsp(L)$ of a link $L$ is the minimal number of crossing changes needed to turn $L$ into a split union of knots. We describe conditions under which certain $\mathbb{R}$-valued link invariants give lower bounds on $wsp(L)$. This result is used both to obtain new bounds on $wsp(L)$ in terms of t…

We prove that every embedding of $K_{2n+1,2n+1}$ into $\R^3$ contains a non-split link of $n$-components. Further, given an embedding of $K_{2n+1,2n+1}$ in $\R^3$, every edge of $K_{2n+1,2n+1}$ is contained in a non-split $n$-component link in $K_{2n+1,2n+1}$.

We show that every p-fold strictly-cyclic branched covering of a b-bridge link in the 3-sphere admits a p-symmetric Heegaard splitting of genus g=(b-1)(p-1). This gives a complete converse to a result of Birman and Hilden, and gives an intrinsic characterization of p-symmetric Heegaard splittings as p-fold strictly-cyc…

In a recent work "Arc-presentation of links: Monotonic simplification" Ivan Dynnikov showed that each rectangular diagram of the unknot, composite link, or split link can be monotonically simplified into a trivial, composite, or split diagram, respectively. The following natural question arises: Is it always possible t…

New homological action on sutured instanton homology defined.

We study intrinsically linked graphs where we require that every embedding of the graph contains not just a non-split link, but a link that satisfies some additional property. Examples of properties we address in this paper are: a two component link with lk(A,L) = k2^r, k not 0, a non-split n-component link where all l…

An immersed concordance between two links is a concordance with possible self-intersections. Given an immersed concordance we construct a smooth four-dimensional cobordism between surgeries on links. By applying $d$-invariant inequalities for this cobordism we obtain inequalities between the $H$-functions of links, whi…

We use the knot homology of Khovanov and Lee to construct link concordance invariants generalizing the Rasmussen $s$-invariant of knots. The relevant invariant for a link is a filtration on a vector space of dimension $2^{|L|}$. The basic properties of the $s$-invariant all extend to the case of links; in particular, a…

We find all 2-Bridge links up to 11 crossings and locate them in Thistlethwaite's link table. The splitting numbers of some links are calculated as a consequence of this identification.

We introduce a version of Khovanov homology for alternating links with marking data, $ω$, inspired by instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology introduced in \cite{KM_unknot} for this marked Khovanov homology collapses on the $E_2$ page fo…

Let $f$ be the gluing map of a Heegaard splitting of a 3-manifold $W$. The goal of this paper is to determine the information about $W$ contained in the image of $f$ under the symplectic representation of the mapping class group. We prove three main results. First, we show that the first homology group of the three man…

In this paper, we give an isotopy classification of 3-bridge spheres of 3-bridge arborescent links, which are not Montesinos links. To this end, we prove a certain refinement of a theorem of J.S. Birman and H.M. Hilden on the relation between bridge presentations of links and Heegaard splittings of 3-manifolds. In the …