Related papers: Concordance to links with an unknotted component
Composition and lattice join (transitive closure of a union) of equivalence relations are operations taking pairs of decidable equivalence relations to relations that are semi-decidable, but not necessarily decidable. This article addresses…
The aim of this survey article is to highlight several notoriously intractable problems about knots and links, as well as to provide a brief discussion of what is known about them.
Building complex software systems necessitates the use of component-based architectures. In theory, of the set of components needed for a design, only some small portion of them are "custom"; the rest are reused or refactored existing…
A gordian unlink is a finite number of unknots that are not topologically linked, each with prescribed length and thickness, and that cannot be disentangled into the trivial link by an isotopy preserving length and thickness throughout. In…
We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a…
The bipolar filtration introduced by T. Cochran, S. Harvey, and P. Horn is a framework for the study of smooth concordance of topologically slice knots and links. It is known that there are topologically slice 1-bipolar knots which are not…
We will prove that, for a $2$ or $3$ component $L$-space link, $HFL^-$ is completely determined by the multi-variable Alexander polynomial of all the sub-links of $L$, as well as the pairwise linking numbers of all the components of $L$. We…
Milnor's $\bar{\mu}$-invariants of links in the $3$-sphere $S^3$ vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in $S^3$. Here we consider knots in thickened surfaces $\Sigma \times [0,1]$,…
Matching Logic is a framework for specifying programming language semantics and reasoning about programs. Its formulas are called patterns and are built with variables, symbols, connectives and quantifiers. A pattern is a combination of…
The multivariate Alexander module of a link L has several subsets that admit quandle operations defined using the module operations. One of them, the fundamental multivariate Alexander quandle, determines the link module sequence of L.
Trivial links are unique up to number of link components, but they can be hard to recognize from arbitrary diagrams. We define a new measure of the complexity of a link embedding, the crumple, and show how this may be used to measure…
We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional…
We construct the complete invariant for fused links. It is proved that the set of equivalence classes of $n$-component fused links is in one-to-one correspondence with the set of elements of the abelization $UVP_n/UVP_n^{\prime}$ up to…
Cochran, Orr, and Teichner developed a filtration of the knot concordance group indexed by half integers called the solvable filtration. Its terms are denoted by $\mathcal{F}_n$. It has been shown that $\mathcal{F}_n/\mathcal{F}_{n.5}$ is a…
A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are…
A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first order differential calculus on such a…
We prove the existence of a smoothly doubly slice, amphicheiral knot with Alexander polynomial 1 and unknotting number 5.
A new model for mechanical computing is demonstrated that requires only two basic parts: links and rotary joints. These basic parts are combined into two main higher level structures: locks and balances, which suffice to create all…
Hom gives an example of a knot with vanishing Upsilon invariant but nonzero epsilon invariant. We build more such knots that are linearly independent in the smooth concordance group.
We introduce the Alexander-Beck module of a knot as a canonical refinement of the classical Alexander module, and we prove that this new invariant is an unknot-detector.