arXiv research

Links with isotopic preimages in $S^3$ are isotopic in $RP^3$.

Given an $m$-component link $L$ in $S^3$ ($m \ge 2$), we construct a family of links which are link homotopic, but not link isotopic, to $L$. Every proper sublink of such a link is link isotopic to the corresponding sublink of $L$. Moreover, if $L$ is an unlink then there exist links that in addition to the above prope…

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

The study finds non-simple isotopy classes of links in 3-manifolds, including Legendrian and pseudo-Legendrian examples.

Infinitely many hyperbolic links in lens space have isotopic lifts in 3-sphere.

New link pairs exist that can't be deformed without increasing length.

Basket links are shown to be isotopic to $T(2,n+1)$.

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

New link invariants derived from crossing multiplexing.

Links can be transformed into many others using a specific operation.

Study of cable links of uniformly thick knots, revealing new isotopy phenomena.

New non-isotopic Seifert surfaces found in 4-ball.

New proof of link factorization theorem, avoiding case exhaustion.

A new Markov theorem for 4D ribbon torus-links.

Study finds non-isotopic transverse tori in Engel manifolds.

We construct infinite families of topologically isotopic but smoothly distinct knotted spheres in many simply connected 4-manifolds that become smoothly isotopic after stabilizing by connected summing with $S^2 \times S^2$, and as a consequence, analogous families of diffeomorphisms and metrics of positive scalar curva…

Finite type invariants separate PL links in 3D space.

Theory of link projections to 3-manifold spines, proving combinatorial moves for isotopic links.

New method to classify simple Smale flows on $S^3$.

The paper links knot properties in projective space to their fundamental groups.

Paper generalizes Yamada polynomial to virtual spatial graphs.

Positive braids have endless filling possibilities.

New measure of knot complexity tied to twist number.

The paper studies knitted surfaces and surface-links, showing their isotopy and closure properties.

All knots are fused isotopic to the unknot via a process known as virtualization. We extend and adapt this process to show that, up to fused isotopy, classical links are classified by their linking numbers.

We give a classification of $n$-component links up to $C_n$-move. In order to prove this classification, we characterize Brunnian links, and have that a Brunnian link is ambient isotopic to a band sum of trivial link and Milnor's links.

Study of strongly invertible Legendrian links in contact 3-space.

Let K be a tunnel number one, fibered link in S^3, with fiber F, and unknotting tunnel $τ$. We show that $τ$ can be isotoped to lie in F.

Paper constructs infinitely many pairs of Seifert surfaces for each link.

The paper explores isotopic triples of triangles in 3D space.

It is shown that two braids represent transversally isotopic links if and only if one can pass from one braid to another by conjugations in braid groups, positive Markov moves, and their inverses.

New link found that can't be smoothly deformed.

Khovanov homology can identify Hopf links.

Curves with constant curvature are flexible and can be deformed.

Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.

The abstract discusses knots and their isotopy to the unknot, proving a specific case.

We show that a transverse link in a contact structure supported by an open book decomposition can be transversely braided. We also generalize Markov's theorem on when the closures of two braids represent (transversely) isotopic links.

Lisa Traynor has described an example of a two-component Legendrian `circular helix link' in the 1-jet space of the circle (with its canonical contact structure) that is topologically but not Legendrian isotopic to that same link with the order of the two components reversed. We give a complete classification of the Le…

This paper studies isotopies of periodic tangles in 3-manifolds using finite covers.

Study equivariant isotopy in higher dimensions, finding exceptions.

Meridian lemma extended to fully alternating links in thickened surfaces.

Affine links in projective space have a specific group property.

New theorem allows transverse links to be braided with rational book structure.

Study shows how to create 2-links with any group in 4-manifolds.

A new classification theorem for links by the authors and Roger Fenn leads to computable link invariants. As an illustration we distinguish the left and right trefoils and recover the result of Carter et al that the 2-twist-spun trefoil is not isotopic to its orientation reverse. We sketch the proof the classification …

An elementary stabilization of a Legendrian link $L$ in the spherical cotangent bundle $ST^*M$ of a surface $M$ is a surgery that results in attaching a handle to $M$ along two discs away from the image in $M$ of the projection of the link $L$. A virtual Legendrian isotopy is a composition of stabilizations, destabiliz…

Paper proves isotopic 2-knots can be transformed by cubulated moves.

We prove that if $L$ is a non-trivial alternating link embedded (without crossings) in a closed surface $F\subset S^3$, then $F$ has a compressing disk whose boundary intersects $L$ in no more than two points. Moreover, whenever the surface is incompressible and $\partial$-incompressible in the link exterior, it can be…