arXiv research

New method for simplifying knots with specific properties.

New knot invariant λ bounds rational unknotting.

New findings on prime theta-curves with simple tangles.

The paper calculates bounds for unknotting rational tangles using knot Floer homology.

We use the rational Witt class of a knot in the 3-sphere as a tool for addressing questions about its unknotting number. We apply these tools to several low crossing knots (151 knots with 11 crossing and 100 knots with 12 crossings) and to the family of n-stranded pretzel knots for various values of n>2. In many cases …

We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numb…

The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.

Introducing a way to modify knots using $n$-trivial rational tangles, we show that knots with given values of Vassiliev invariants of bounded degree can have arbitrary unknotting number (extending a recent result of Ohyama, Taniyama and Yamada). The same result is shown for 4-genera and finite reductions of the homolog…

This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we prove theorems characterizing when the numerator of the sum of two rational tangles is…

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…

Knots without 2-torsion have minimal Khovanov homology rank.

In the note we study Legendrian and transverse knots in rationally null-homologous knot types. In particular we generalize the standard definitions of self-linking number, Thurston-Bennequin invariant and rotation number. We then prove a version of Bennequin's inequality for these knots and classify precisely when the …

Study contact structures on lens spaces, classifying rational knots.

We consider the recently introduced knotting-unknotting game, in which two players take turns resolving crossings in a knot diagram which initially is missing all its crossing information. Once the knot is fully resolved, the winner is decided by whether the knot is equivalent to the unknot. In this paper we determine …

We classify Legendrian rational unknots with tight complements in the lens spaces L(p,1) up to coarse equivalence. As an example of the general case, this classification is also worked out for L(5,2). The knots are described explicitly in a contact surgery diagram of the corresponding lens space.

Computer experiments reveal complex knots that don't simplify.

We examine certain symmetries in the deficiencies of a rational surgery on a knot in $S^3$ by comparing the $\text{Spin}^c$-structures on the rational surgery with those on a related integral surgery. We then provide an application of these symmetries in the form of a theorem that obstructs Dehn surgeries in $S^3$. Thi…

Computing unlinking number is usually very difficult and complex problem, therefore we define BJ-unlinking number and recall Bernhard-Jablan conjecture stating that the classical unknotting/unlinking number is equal to the BJ-unlinking number. We compute BJ-unlinking number for various families of knots and links for w…

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…

Boring is an operation which converts a knot or two-component link in a 3--manifold into another knot or two-component link. It generalizes rational tangle replacement and can be described as a type of 2--handle attachment. Sutured manifold theory is used to study the existence of essential spheres and planar surfaces …

Study contact structures on specific manifolds, proving infinite non-isotopic structures and tight/overtwisted classifications.

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…

Study tangle equations linking enzyme actions to knot theory.

Shows large unknotting number for simple knots.

Spatial embeddings of planar graphs can have higher unknotting numbers than crossing numbers.

Agent finds unknotting sequences for complex knots.

Paper shows that for torus knots, the pinch number equals the unoriented band unknotting number.

New Boolean algebra method shows knot unknotting number is (c+1)/2.

Lower bounds on unknotting number for cabled knots.

Determining unknotting numbers is a large and widely studied problem. We consider the more general question of the unknotting number of a spatial graph. We show the unknotting number of spatial graphs is subadditive. Let $g$ be an embedding of a planar graph $G$, then we show $u(g) \geq \max\{u(s) |$ $s$ is a non-overl…

The unknotting number is the classical invariant of a knot. However, its determination is difficult in general. To obtain the unknotting number from definition one has to investigate all possible diagrams of the knot. We tried to show the unknotting number can be obtained from any one diagram of the knot. To do this we…

This paper concerns the H(2)-unknotting numbers of links related to 2-bridge links. It consists of three parts. In the first part, we consider a necessary and sufficient condition for a 2-bridge link to have H(2)-unknotting number one. The second part concerns an explicit form of composite links with H(2)-unknotting nu…

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…

New measure shows how links can be untangled as twists increase.

Grid homology shows knot unknotting lower bound.

Study on bounds of knot untangling for specific types of knots.

Study determines $Δ$-unknotting numbers for two-bridge knots.

The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot obtained from the standard (p,q)-torus knot diagram by replacing all crossings o…

The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. The algebraic unknotting number is the minimum number of crossing changes needed to transform a knot into an Alexander polynomial-one knot. We work with a generalization of unknotting number due to Math…

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

New bounds and examples for sphere unknotting numbers.

Study on knot properties, showing relation between unknotting and crossing numbers.

Knot 11n102 requires 2 changes to untangle.

Study shows quasi-additivity of Δ-unknotting number for braids.

RL pipeline simplifies knot diagrams, including very hard unknots.

We use Heegaard Floer homology to obtain bounds on unknotting numbers. This is a generalisation of Ozsvath and Szabo's obstruction to unknotting number one. We determine the unknotting numbers of 9_10, 9_13, 9_35, 9_38, 10_53, 10_101 and 10_120; this completes the table of unknotting numbers for prime knots with crossi…

The paper calculates the Δ-unknotting number for positive pretzel knots.

New number bounds knot complexity, including unknotting and crosscap numbers.