arXiv research

New links found that aren't concordant to any with a single unknotted component.

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…

The paper finds virtual knots with specific unknotting indices.

We prove the perhaps surprising result that given any three polygonal unknots in $\R^3$, then we may form the Borromean rings out of them through rigid motions of $\R^3$ applied to the individual components together with possible scaling of the components. We also prove that if at least two of the unknots are planar, t…

The paper examines surgeries on 2-component L-space links and identifies key properties of the components.

It is shown that any closed three-manifold M obtained by integral surgery on a knot in the three-sphere can always be constructed from integral surgeries on a 3-component link L with each component being an unknot in the three-sphere. It is also interesting to notice that infinitely many different integral surgeries on…

We describe a method for generating minimal hard prime surface-link diagrams. We extend the known examples of minimal hard prime classical unknot and unlink diagrams up to three components and generate figures of all minimal hard prime surface-unknot and surface-unlink diagrams with prime base surface components up to …

Verifies knot conjecture for 24-crossing knots.

Using unknotting number, we introduce a link diagram invariant of Hass and Nowik type, which changes at most by 2 under a Reidemeister move. As an application, we show that a certain infinite sequence of diagrams of the trivial two-component link need quadratic number of Reidemeister moves for being unknotted with resp…

The surgery unknotting number of a Legendrian link is defined as the minimal number of particular oriented surgeries that are required to convert the link into a Legendrian unknot. Lower bounds for the surgery unknotting number are given in terms of classical invariants of the Legendrian link. The surgery unknotting nu…

New unknots with geometric constraints exist, proving a long-standing conjecture.

Combinatorial two-player games have recently been applied to knot theory. Examples of this include the Knotting-Unknotting Game and the Region Unknotting Game, both of which are played on knot shadows. These are turn-based games played by two players, where each player has a separate goal to achieve in order to win the…

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

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.

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

Theory is developed for linear-quadratic at infinity generating families for Legendrian knots in R^3. It is shown that the unknot with maximal Thurston--Bennequin invariant of -1 has a unique linear-quadratic at infinity generating family, up to fiber-preserving diffeomorphism and stabilization. From this, invariant ge…

The paper classifies virtual links using the arc shift operation.

We construct and prove a diagrammatic version of the Duflo isomorphism between the invariant subalgebra of the symmetric algebra of a Lie algebra and the center of the universal enveloping algebra. This version implies the original for metrized Lie algebras (Lie algebras with an invariant non-degenerate bilinear form).…

In this paper we compute the sharp lower bounds for the crossing number of $n$-string $k$-loop essential tangles. For essential tangles with only string components, we characterise the ones with the minimum crossing number for a given number of components, both when the tangle has knotted strings or only unknotted stri…

The paper studies unlinking links with minimal crossing changes.

We show that the following unlinking strategy does not always yield an optimal sequence of crossing changes: first split the link with the minimal number of crossing changes, and then unknot the resulting components.

Study on folded ribbon knots and their minimum length.

Smooth tori in S^4 are topologically unknotted.

For every odd integer $c\ge 21$, we raise an example of a prime component-preservingly amphicheiral link with the minimal crossing number $c$. The link has two components, and consists of an unknot and a knot which is $(-)$-amphicheiral with odd minimal crossing number. We call the latter knot a {\it Stoimenow knot}. W…

We classify Legendrian unknots in overtwisted contact structures on $S^3$. In particular, we show that up to contact isotopy for every pair $(n,\pm(n-1))$ with $n>0$ there are exactly two oriented non-loose Legendrian unknots in $S^3$ with Thurston-Bennequin invariant $n$ and rotation number $\pm(n-1)$. (Only one overt…

In a recent work of Ayaka Shimizu$^{[5]}$, she defined an operation named region crossing change on link diagrams, and showed that region crossing change is an unknotting operation for knot diagrams. In this paper, we prove that region crossing change on a 2-component link diagram is an unknotting operation if and only…

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…

Classifies links with minimal Khovanov homology rank.

We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard, including detecting whether a link contains a trivial sublink with $n$ componen…

New methods detect and create Brunnian links efficiently.

Study extends knot genus results to two-component alternating links.

Three disjoint rays in euclidean 3-space form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.

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…

New proof for minimizing tunnel systems in satellite chain links.

Let P be a knot in an unknotted solid torus (i.e. a satellite operator or pattern), K a knot in S^3 and P(K) the satellite of K with pattern P. For any satellite operator P, this correspondence gives a function P : C -> C on the set of smooth concordance classes of knots. We give examples of winding number one satellit…

We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.

Study on hard Legendrian unknots using normal rulings.

Agent finds unknotting sequences for complex knots.

Study categorizes knots and links as rigid or shaky based on Reidemeister moves.

The (ordinary) unknotting-number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. It is very natural to consider the `unknotting-number' associated with other local-moves on n-dimensional knots, where n is a natural number. In this paper we prove the follow…

New foliations found in 3D spaces from positive braids.

Shows large unknotting number for simple knots.

New invariant measures how many twists are needed to unknot welded knots.

The paper extends CF-moves to classify virtual links of any number of components.

If L_1 and L_2 are two Brunnian links with all pairwise linking numbers 0, then we show that L_1 and L_2 are equivalent if and only if they have homeomorphic complements. In particular, this holds for all Brunnian links with at least three components. If L_1 is a Brunnian link with all pairwise linking numbers 0, and t…

We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on alternating 3-braid knots is sufficient to identify all unknotting number one alte…

Grid homology shows knot unknotting lower bound.

RL pipeline simplifies knot diagrams, including very hard unknots.