A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Infinitely many splitting spheres found for unlinked 2-spheres in 4-space.
New spanning 3-disks found for unlink in 4-sphere.
Study sharpens unlinking number bounds for special alternating links.
New rational band moves simplify knot classification.
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.
The paper offers new methods to determine if certain 3D links can be formed by intersecting spheres in 4D space.
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 …
In this paper, we study surfaces embedded in -manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary -manifold. This extends work of Swenton and Kearton-Kurlin in . As an application, we show that bridge trisections of isotopic surfaces in a trisected …
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.
New spheres can split a 4D link in ways not possible in 3D.
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 -graded vector space structure of the HOMFLYPT homology over detects unlinks. Our proof relies on a theorem of Batson and Seed stating that the -graded vector space structure of the Khovanov homology over $\mathbb{Z}_2…
We prove a homological stability theorem for unlinked circles in -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.
We compute the group of link homotopy classes of link maps of two 2-spheres into 4-space. It turns out to be free abelian, generated by geometric constructions applied to the Fenn-Rolfsen link map and detected by two self-intersection invariants introduced by Paul Kirk in this setting. As a corollary, we show that any …
Study surfaces in 4-manifolds using banded unlink diagrams.
We prove that deciding if a diagram of the unknot can be untangled using at most Riedemeister moves (where is part of the input) is NP-hard. We also prove that several natural questions regarding links in the -sphere are NP-hard, including detecting whether a link contains a trivial sublink with componen…
A path integral on a link complement of a three-sphere fixes a vector (the "link state") in Chern-Simons theory. The link state can be written in a certain basis with the colored link invariants as its coefficients. We use symmetric webs to systematically compute the colored link invariants, by which we can write down …
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.
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 , finding a -component unlink as a sub…
Alexander invariant created for doodles, vanishes on unlinked doodles.
Knot invariants and quiver stability linked through full twists.
Given a link map f into a manifold of the form Q = N \times \Bbb R, when can it be deformed to an unlinked position (in some sense, e.g. where its components map to disjoint \Bbb R-levels) ? Using the language of normal bordism theory as well as the path space approach of Hatcher and Quinn we define obstructions \widet…
Enhances Hantzsche's theorem for 3-manifolds in 4D.
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.
A knot K in the 3-sphere is said to have Property nR if, whenever K is a component of an n-component link L and some integral surgery on L produces the connected sum of n copies of S^1 x S^2, there is a sequence of handle slides on L that converts L into a 0-framed unlink. The Generalized Property R Conjecture is that …
Paper constructs a spectral sequence linking annular Khovanov homology to reduced Khovanov homology.
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…
New method for quandle presentations of surface knots in 4-manifolds.
Study categorizes knots and links as rigid or shaky based on Reidemeister moves.
We study a subset of square free positive braids and we give a few algebraic characterizations of them and one geometric characterization: the set of positive braids whose closures are unlinks. We describe canonical forms of these braids and of their conjugacy classes.
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…
The depth of a link measures the minimum height of a resolving tree for the link whose leaves are all unlinks. We show that the depth of the closure of a strictly positive braid word is the length of the word minus the number of distinct letters.
Proves links can be simplified to trivial form in few changes, limiting Milnor's invariants.
Let L \subset S^3 denote an alternating link and Sigma(L) its branched double-cover. We give a short proof of the fact that the fundamental group of Sigma(L) admits a left-ordering iff L is an unlink. This result is originally due to Boyer-Gordon-Watson.
It is proved that every disconnected surface-link with meridian-based free fundamental group is a trivial (i.e., an unknotted-unlinked) surface-link. This result is a surface-link version of the author's recent announcement result on smooth unknotting of a surface-knot.