arXiv research

Enhances Hantzsche's theorem for 3-manifolds in 4D.

The paper offers new methods to determine if certain 3D links can be formed by intersecting spheres in 4D space.

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…

Paper finds first examples of unlinked knots that can't be separated.

New spanning 3-disks found for unlink in 4-sphere.

The paper classifies twisted torus links that are unlinks.

Delta-unlinking number measures how to unlink algebraically split links.

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 …

Infinitely many splitting spheres found for unlinked 2-spheres in 4-space.

Study sharpens unlinking number bounds for special alternating links.

In this paper we investigate the unlinking numbers of 10-crossing links. We make use of various link invariants and explore their behaviour when crossings are changed. The methods we describe have been used previously to compute unlinking numbers of links with crossing number at most 9. Ultimately, we find the unlinkin…

Simplified plat diagrams for unlink without stabilization.

In links with two components there are three different types of crossings: self-crossings in the first component, self crossings in the second component, and crossings between components. In this paper we examine the minimum number of crossing changes needed to unlink without changing the crossings between components. …

New examples of gordian unlinks show different rope geometries.

The paper examines when 2-string tangles can be embedded into specific link types.

We apply the Rasmussen spectral sequence to prove that the $\mathbb{Z}^3$-graded vector space structure of the HOMFLYPT homology over $\mathbb{Z}_2$ detects unlinks. Our proof relies on a theorem of Batson and Seed stating that the $\mathbb{Z}^2$-graded vector space structure of the Khovanov homology over $\mathbb{Z}_2…

We prove a homological stability theorem for unlinked circles in $3$-manifolds and give an application to certain groups of diffeomorphisms of 3-manifolds.

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.

Let n be a positive integer. We provide a Khovanov homology proof of the following classical fact: If the closure of an n-strand braid is the n-component unlink, then the braid is trivial.

Study shows torsion order bounds band-unlinking number for knot cobordisms.

Study surfaces in 4-manifolds using banded unlink diagrams.

Minimal moves for surfaces in 4D identified.

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…

We generalise theorems of Cochran-Lickorish and Owens-Strle to the case of links with more than one component. This enables the use of linking forms on double branched covers, Heegaard Floer correction terms, and Donaldson's diagonalisation theorem to complete the table of unlinking numbers for nonsplit prime links wit…

We prove that if n\ge1, then an (n+1)-component Brunnian link L in a connected, oriented 3-manifold is C_n-equivalent to an unlink. We also prove that if n\ge2, then L can not be distinguished from an unlink by any Goussarov-Vassiliev finite type invariant of degree<2n.

New rational band moves simplify knot classification.

We prove that certain problems naturally arising in knot theory are NP--hard or NP--complete. These are the problems of obtaining one diagram from another one of a link in a bounded number of Reidemeister moves, determining whether a link has an unlinking or splitting number $k$, finding a $k$-component unlink as a sub…

Alexander invariant created for doodles, vanishes on unlinked doodles.

Knot invariants and quiver stability linked through full twists.

We show that every quasipositive link has a quasipositive minimal braid representative, partially resolving a question posed by Orevkov. These quasipositive minimal braids are used to show that the maximal self-linking number of a quasipositive link is bounded below by the negative of the minimal braid index, with equa…

New method for regression in high-dimensional space using mixture modeling and optimal transport.

Paper constructs a spectral sequence linking annular Khovanov homology to reduced Khovanov homology.

The paper examines Einstein doubly warped product manifolds with a semi-symmetric metric connection.

Characterizes spacetimes using doubly torqued vectors.

New invariant measures doubly slice links, disproving previous bounds.

Characterizes a specific type of spacetime using vector fields.

Characterizes and examines gradient solitons on doubly warped product manifolds.

The article enumerates doubly symmetric diagrams for knots up to 18 crossings.

We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length < k are all zero into a link with vanishing Milnor invariants of length < 2k, and we provide formulas for th…

Proposes DR-ACI for causal effect intervals with temporal dependence.

Simplified tutorial on doubly robust learning for causal inference.

We show that if the connected sum of two knots with coprime Alexander polynomials is doubly slice, then the Ozsváth-Szabó correction terms as smooth double sliceness obstructions vanish for both knots. Recently, Jeffrey Meier gave smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice. …

In this article, we present a complete study of two disjoint classes of conformal vector fields on doubly warped product manifolds as well as on doubly warped space-times. Then we study Ricci solitons on doubly warped product manifollds admitting these types of conformal vector fields.

Identifies doubly slice genera for 2909 prime knots with up to 12 crossings.

The paper shows some Montesinos links can't be doubly sliced strongly.

We define an obstruction for a knot to be Z[Z]-homology ribbon, and use this to provide restrictions on the integers that can occur as the triple linking numbers of derivative links of knots that are either homotopy ribbon or doubly slice. Our main application finds new non-doubly slice knots. In particular this gives …

In this paper, we study surfaces embedded in $4$-manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary $4$-manifold. This extends work of Swenton and Kearton-Kurlin in $S^4$. As an application, we show that bridge trisections of isotopic surfaces in a trisected …

Hamiltonian cycles found in toroidal maps.