arXiv research

New 2-links created in 4D spaces, topologically unknotted but not smoothly.

Topologically protected vortex knots and links are proposed and proven.

New method detects essential tori in mixed singularity links.

New colored link invariants using multi-quandles.

Invariants for colored links found, with topological protection of certain knots.

To every oriented link $L$, we associate a topologically defined biquandle $\widehat{\mathcal{B}}_{L}$, which we call the topological biquandle of $L$. The construction of $\widehat{\mathcal{B}}_{L}$ is similar to the topological description of the fundamental quandle given by Matveev. We find a presentation of the top…

We generalize an algorithm of Rudolph to establish that every link is topologically concordant to a strongly quasipositive link.

The recently suggested tangle calculus for knot polynomials is intimately related to topological string considerations and can help to build the HOMFLY-PT invariants from the topological vertices. We discuss this interplay in the simplest example of the Hopf link and link $L_{8n8}$. It turns out that the resolved conif…

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 2-bipolar. For knots, this is the highest known level at which the filtration do…

Link homology theories connect to 4-manifold invariants and TQFTs.

We give infinitely many $2$-component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any $2$-component link with trivial Alexander polynomial. Our examples are pairwise non-concordant.

We use an estimate on the Thurston--Bennequin invariant of a Legendrian link in terms of its Kauffman-polynomial to show that links of topological unknots, e.g. the Borromean rings or the Whithead link, may not be represented by Legendrian links of Legendrian unknots.

New Milnor's invariant condition for topologically slice links.

We introduce a notion of topological quandle. Given a topological quandle $Q$ we associate to every classical link $L$ in $\R ^3$ an invariant $J_Q(L)$ which is a topological space (defined up to a homeomorphism). The space $J_Q(L)$ can be interpreted as a space of colourings of a diagram of the link $L$ with colours f…

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

Topological model created for HOMFLY-PT polynomial from link diagrams.

Polynomial algorithm found for alternating link equivalence.

New methods show hyperbolicity of Brunnian links.

Study topological Iwasawa invariants for 3-sphere links, proving density results.

Computes link invariants in real projective 3-space using topological vertex.

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 $g_4(\ell_i)=i$.

We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply (1+1)-dimensional unoriented topological quantum field theories to Bar-Natan's geometric formalism to define new theories for stable equivalence classes.

We give new examples of 2-component links with linking number one and unknotted components that are topologically concordant to the positive Hopf link, but not smoothly so - in fact they are not smoothly concordant to the positive Hopf link with a knot tied in the first component. Such examples were previously construc…

Finite type invariants separate PL links in 3D space.

It was shown by Jim Davis that a 2-component link with Alexander polynomial one is topologically concordant to the Hopf link. In this paper, we show that there is a 2-component link with Alexander polynomial one that has unknotted components and is not smoothly concordant to the Hopf link, answering a question of Jim D…

Study on packing links with geometric constraints.

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…

Paper introduces simplified formulas for Milnor's triple linking number.

New method analyzes knots and links using multiscale Gauss link integral.

Extends Milnor's invariants to knots and links in 3-manifolds.

We give new Casson-Gordon style obstructions for a two-component link to be topologically concordant to the Hopf link.

We construct infinitely many Legendrian links in the standard contact $\mathbb{R}^3$ with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in $S^3$ that bound topologically distinct pieces of algebraic curves in $B^4 \subset \mathbb{C}^2$, is applied to find contact 3-…

There is a topological embedding $ι:\mathbb{S}^1\to\mathbb{R}^5$ such that $π_3(\mathbb{R}^5\setminusι(\mathbb{S}^1))=0$. Therefore, no $3$-sphere can be linked with $ι(\mathbb{S}^1)$.

Proves a plumbing-multiplicative property of a Links-Gould invariant.

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

Paper links set derivatives to its orthogonal projections.

We show that there are links whose individual components are concordant to the unknot, but which are not concordant to any link with unknotted components. We give examples in the topological category, and examples in the smooth category which are topologically slice. We also give generalizations regarding components of…

Study on Alexander polynomials in braids, linking number theory and topology.

New mathematical tools for studying knots and links.

New link topology connects permutation discrepancies to Diaconis-Graham inequalities.

Four-dimensional surgery is used to show that a two component link with Alexander polynomial one is topologically concordant to the Hopf link.

Stable knots and links can exist in electromagnetic fields.

Study on homology groups of cDV singularity links, identifying their topology.

Extends topological results for nonpositive curvature spaces.

PHLP uses persistent homology to interpret graph link prediction.

In this paper we define and study flexible links and flexible isotopy in projective space. Flexible links are meant to capture the topological properties of real algebraic links. We classify all flexible links up to flexible isotopy using Ekholms interpretation of Viros encomplexed writhe.

Starfield optimizes satellite links for LEO mega-constellations by aligning ISLs with regional traffic patterns.

The study examines the topology of map germs and their images.