Related papers: Singular link concordance implies link homotopy in…
We show that every non-trivial strongly quasipositive link is smoothly concordant to infinitely many pairwise non-isotopic strongly quasipositive links. In contrast to our result, Baker conjectured that smoothly concordant strongly…
In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse…
A theory of signatures for odd-dimensional links in rational homology spheres is studied via their generalized Seifert surfaces. The jump functions of signatures are shown invariant under appropriately generalized concordance and a special…
We give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of ` symmetric' links in $S^3$.
Asaeda-Przytycki-Sikora, Manturov, and Gabrov\v{s}ek extended Khovanov homology to links in $\mathbb{RP}^3$. We construct a Lee-type deformation of their theory, and use it to define an analogue of Rasmussen's s-invariant in this setting.…
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…
This paper is a generalization of the author's previous work on link homotopy to link concordance. We show that the only real-valued finite type link concordance invariants are the linking numbers of the components.
Links in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeister-type moves on diagrams connecting…
Configuration space integrals have in recent years been used for studying the cohomology of spaces of (string) knots and links in $\mathbb{R}^n$ for $n>3$ since they provide a map from a certain differential algebra of diagrams to the…
We consider a possibility of the existence of intersection homology morphism, which would be associated to a map of analytic varieties. We assume that the map is an inclusion of codimension one. Then the existence of a morphism follows from…
We present a new short proof of the explicit formula for the group of links (and also link maps) in the 'quadruple point free' dimension. Denote by $L^m_{p,q}$ (respectively, $C^{m-p}_p$) the group of smooth embeddings $S^p\sqcup S^q\to…
The theory of signature invariants of links in rational homology spheres is applied to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, an explicit formula is derived to compute…
We generalize an algorithm of Rudolph to establish that every link is topologically concordant to a strongly quasipositive link.
We show that the isomorphism of 3-connected planar graphs can be decided in deterministic log-space. This improves the previously known bound UL$\cap$coUL of Thierauf and Wagner.
We prove the following "linkage" theorem: two p-regular graphs of the same genus can be obtained from one another by a finite alternating sequence of one-edge-contractions; moreover this preserves 3-edge-connectivity. We use the linkage…
We consider a canonical $S^1$ action on $S^3$ which is defined by $(\rho,(z_1,z_2))\mapsto (z_1\rho^p,z_2\rho^q)$ for $\rho\in S^1$ and $(z_1,z_2)\in S^3\subset {\mathbb C}^2$. We consider a link consisting of finite orbits of this action,…
To an oriented link in a solid torus we associate a trace graph in a thickened torus in such a way that links are isotopic if and only if their trace graphs can be related by moves of finitely many standard types. The key ingredient is a…
We show that there are infinitely many triples of non-isotopic hyperbolic links in the lens space $L(4,1)$ such that the three lifts of each triple in $S^{3}$ are isotopic. They are obtained as the lifts of links in $S^{3} / Q_{8}$ by…
Two links are called link-homotopic if they are transformed to each other by a sequence of self-crossing changes and ambient isotopies. The notion of link-homotopy is generalized to spatial graphs and it is called component-homotopy. The…
Link equivalence up to isotopy in a 3-space is the problem that lies at the root of knot theory, and is important in 3-dimensional topology and geometry. We consider its restriction to alternating links, given by two alternating diagrams…