The paper explores Gordian complexes of knots and virtual knots using region crossing changes and arc shift moves.
problem Defining and analyzing Gordian complexes of knots and virtual knots using specific local moves.
method Region crossing change and arc shift move to construct Gordian complexes.
result Existence of arbitrarily high dimensional simplices in both Gordian complexes.
We introduce new simplicial complexes by using various invariants and local moves for knots, which give generalizations of the Gordian complex defined by Hirasawa and Uchida. In particular, we focus on the simplicial complex defined by using the Alexander-Conway polynomial and the Delta-move, and show that the simplici…
A knot K is called Gordian adjacent to a knot L if there exists an unknotting sequence for L containing K. We provide a sufficient condition for Gordian adjacency of torus knots via the study of knots in the thickened torus. We also completely describe Gordian adjacency for torus knots of index 2 and 3 using Levine-Tri…
Paper finds first examples of unlinked knots that can't be separated.
problem Separating knots without changing their length and thickness.
method Constructs infinite families of 2-component gordian unlinks and n-component links for n≥2. result Found infinite families of 2-component gordian unlinks that cannot be separated.
Quotients of Gordian and H(2)-Gordian graphs are hyperbolic.
problem Investigate quotients of Gordian and H(2)-Gordian graphs under knot invariants.
method Defined equivalence relations by knot invariants (det, Jones span, tricolorability) and showed quotient graphs are Gromov hyperbolic.
result Quotients of H(2)-Gordian graph of links modulo span of Jones polynomial is isomorphic to complete graph.
New invariant measures how many twists are needed to unknot welded knots.
problem Measuring complexity of welded knots.
method Local twist move, Alexander quandle coloring, Gordian distance.
result Established lower bound on twist number and related it to other knot invariants.
Lower bounds on Gordian distance using Blanchfield pairings.
problem Calculating the Gordian distance between knots.
method Using Blanchfield pairings to establish lower bounds.
result At least 195 pairs of knots have a Gordian distance of 3.
Using Blanchfield pairings, we show that two Alexander polynomials cannot be realized by a pair of matrices with Gordian distance one if a corresponding quadratic equation does not have an integer solution. We also give an example of how our results help in calculating the Gordian distances, algebraic Gordian distances…
The study finds conditions for positive braid knots to be Gordian adjacent and explores their unknotting sequences.
problem Understanding Gordian adjacency in positive braid knots.
method Manipulating braid words to find conditions for Gordian adjacency and exploring unknotting sequences.
result There are only finitely many positive braid knots for a given unknotting number.
New bounds for knot distances using Khovanov homology.
problem Calculating precise distances between knots.
method Using Khovanov homology to refine existing bounds.
result Improved bounds for Gordian distances of knots.
The Gordian distance between two knots measures how many crossing changes are needed to transform one knot into the other. It is known that there are always infinitely many non-equivalent knots `between' a pair of knots of Gordian distance two. In this paper we prove an extreme generalisation of this fact: there are kn…
Study of Gordian graphs' behavior at infinity for various local moves.
problem Behavior of Gordian graphs at infinity for different local moves.
method Analysis of unbounded connected components of complements of bounded subsets and finite subsets.
result Complete description of Gordian graphs' behavior at infinity for three families of local moves.
We give lower bounds for the Gordian distance and the unknotting number of handlebody-knots by using Alexander biquandle colorings. We construct handlebody-knots with Gordian distance n and unknotting number n for any positive integer n.
New examples of gordian unlinks show different rope geometries.
problem Tackling the existence of non-trivial unknots with prescribed length and thickness.
method Provided the first examples of gordian unlinks with thickness in [1,2).
result Thinner normal tubes lead to different rope geometries.
Study knots with genus one, finds Gordian distance and cosmetic crossing constraints.
problem Understanding knots with genus one and their properties.
method Using HOMFLT polynomials to find obstructions for Gordian distance and cosmetic crossings.
result Proves the (generalized) cosmetic crossing conjecture for genus one pretzel knots.
A site-specific Gordian distance between two spatial embeddings of an abstract graph is the minimal number of crossing changes from one to another where each crossing change is performed between two previously specified abstract edges of the graph. It is infinite in some cases. We determine the site-specific Gordian di…
Starting from a divide, i.e. a generic immersion of finitely many copies of the interval [0,1] in the disk, we construct a classical link in the 3-sphere. We prove that the link's complement fibers over the circle, if the divide is connected. Moreover, we compute the monodromy diffeomorphism from the combinatorics of t…
Paper introduces a new invariant for planar knotoids.
problem Defining an invariant for planar knotoids.
method Using Gauss diagrams and transcendental functions.
result The invariant is a Vassiliev invariant of order one.
We present a class of knots associated with labelled generic immersions of intervals into the plane and compute their Gordian numbers and 4-dimensional invariants. At least 10% of the knots in Rolfsen's table belong to this class of knots. We call them track knots. They are contained in the class of quasipositive knots…
We define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The n-th subgraph of this family is generated by all knots whose fundamental groups surject to a symmetric group with parameter at least n, where all meridians are mapped to transpositions. Incidentally, we verify the Meridio…
Given a link in S3 we will use invariants derived from the Alexander module and the Blanchfield pairing to obtain lower bounds on the Gordian distance between links, the unlinking number and various splitting numbers. These lower bounds generalise results recently obtained by Kawauchi. We give an application restric…
Defines a new Upsilon torsion function for knot Floer homology.
problem Obtaining constraints on knot cobordisms.
method Defines a one-parameter family of Heegaard Floer torsion invariants.
result Provides new obstructions related to the Gordian distance between knots.
We show that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrarily high bridge distance.
New knot graphs show most are not Gromov hyperbolic, with special cases.
problem Characterizing Gromov hyperbolicity in knot graphs.
method Defining knot graphs and proving non-hyperbolicity.
result Most knot graphs are not Gromov hyperbolic, with exceptions.
New link pairs exist that can't be deformed without increasing length.
problem Link isotopy and thick isotopy constraints.
method Constructed isotopic link pairs with preserved total length.
result Link pairs exist that are not thick isotopic.
Unified proof of knot unknotting bounds using Ma-Qiu index.
problem Finding bounds on the number of moves to unknot knots.
method Using the Ma-Qiu index to bound presentation distances and Gordian distances.
result Unified proof of various unknotting number bounds.
Study on knot unknotting numbers and their behavior under connected sums.
problem Behavior of knot unknotting numbers under connected sums.
method Analyzing the band-unknotting number and its sub-additivity properties.
result Infinitely many examples showing unb(K1#K2)<unb(K1)+unb(K2) and unb(K1#K2)<unb(Ki) for i=1,2. We study the gordian graph of all knots in R3: two knots are adjacent if they differ by a single crossing change. We prove that this graph contains isometrically an infinite countable tree with infinite valency, and that the complement of any finite subset is connected.
New unknots with geometric constraints exist, proving a long-standing conjecture.
problem Existence of distinct isotopy classes of physical unknots with geometric constraints.
method Parametrised thickness and geometric thresholds to fragment isotopy classes.
result Existence of gordian unknots with prescribed geometric constraints.
Algorithm simplifies Khovanov homology computations for 4-strand torus links.
problem Computing Khovanov homology of 4-strand torus links.
method Algorithmic Morse theoretic simplifications postponed until the end.
result Non-trivial Khovanov homology groups in all homological degrees for 4-strand torus links.
Physical knots and links are one-dimensional submanifolds of R^3 with fixed length and thickness. We show that isotopy classes in this category can differ from those of classical knot and link theory. In particular we exhibit a Gordian Split Link, a two component link that is split in the classical theory but cannot be…
We describe two locally finite graphs naturally associated to each knot type K, called Reidemeister graphs. We determine several local and global properties of these graphs and prove that in one case the graph-isomorphism type is a complete knot invariant up to mirroring. Lastly, we introduce another object, relating t…
New lower bounds on the unknotting number of a knot are constructed from the classical knot signature function. These bounds can be twice as strong as previously known signature bounds. They can also be stronger than known bounds arising from Heegaard Floer and Khovanov homology. Results include new bounds on the Gordi…
New framework distinguishes knots via neighborhood invariants.
problem Distinguishing knots and knotoids.
method Study of knotoid spectra and neighborhood invariants.
result Neighborhood invariants can distinguish knots of higher Gordian distance.
Knot Floer homology provides bounds on knot unknotting numbers.
problem Bounding the unknotting number of knots.
method Using knot Floer homology to construct invariants l^-(K), l^+(K), and l(K).
result The invariant l(K) only vanishes for the unknot, and gives lower bounds on the unknotting number.
The paper extends graph signatures to Klein graphs and foams, linking signatures to knot properties.
problem Extending graph signatures to Klein graphs and foams.
method Developed an analogy of Murasugi's bounds and used signatures to lower bound knot properties.
result Lower bounds on negative orbifold Euler characteristics and unknotting numbers.
Geometric trick simplifies link homotopy and concordance.
problem Homotopy and concordance of links in homology spheres.
method Relative Whitney trick to remove double points.
result Links in homology spheres can be simplified to topologically slice links.
We study three knot invariants related to smoothly immersed disks in the four-ball. These are the four-ball crossing number, which is the minimal number of normal double points of such a disk bounded by a given knot; the slicing number, which is the minimal number of crossing changes to a slice knot; and the concordanc…
The study eliminates infinite families of knots with nontrivial Alexander polynomials and improves unknotting number data.
problem Identifying knots with nontrivial Alexander polynomials and improving knot classification.
method Elimination of infinite families of knots and use of determinants to improve unknotting number data.
result Elimination of infinite families of knots with nontrivial Alexander polynomials and improvement of unknotting number data.
New methods for delta-moves on algebraically split links identified.
problem Understanding delta-moves on algebraically split links.
method Introducing self and mixed delta-moves, proving equivalence, and calculating delta-splitting numbers.
result Two links are mixed delta-equivalent if they have the same pairwise linking number and components.
New examples show limits of physical link isotopies.
problem Limits of physical link isotopies in thick and thickly embedded links.
method Construction of specific examples of thick and thickly embedded links.
result Explicit examples of links that cannot be split through thick homotopies.
Proves links can be simplified to trivial form in few changes, limiting Milnor's invariants.
problem Limits of Milnor's invariants in understanding unlinking number.
method Sequence of links, crossing changes, and geometric filtration.
result Crossing changes to Ck-trivial link grows quadratically with number of components. New condition prevents hyperbolic spaces from matching curve complexes.
problem Identifying when hyperbolic spaces cannot match curve complexes.
method Analyzing specific hyperbolic complexes and identifying a condition.
result Identified a condition preventing quasi-isometry between hyperbolic spaces and curve complexes.
Study on complex line fields on almost-complex manifolds, proving existence conditions.
problem Existence of linearly independent complex line fields on almost-complex manifolds.
method Prove necessary and sufficient conditions for the existence of one, two, or three fields over certain manifolds.
result Necessary and sufficient condition for the existence of complex line fields over certain manifolds.
Homotopy types of curve and arc complexes are studied.
problem Understanding the homotopy types of curve and arc complexes.
method Proving homotopy equivalence and contractibility of complexes.
result Fine curve complex is homotopy equivalent to curve complex, fine arc complex is contractible.
This research explores complex-valued neural networks and their implementation.
problem The challenges of implementing complex-valued neural networks and their potential for non-complex data.
method Detailed theory and implementation of CVNN, including Wirtinger calculus, complex backpropagation, and modules like complex layers and activation functions. Python implementation using cvnn toolbox.
result Demonstrates the potential of CVNN for non-complex data through simulations.
Paper introduces fat CW complexes including all closed manifolds.
problem No specific problem stated, focuses on introducing new CW complexes.
method Introduces a new smooth version of CW complexes called fat CW complexes.
result Fat CW complexes include all closed manifolds and have desirable properties.
The paper discusses q-deformations of the Aomoto complex.
problem Deformation of cochain complexes associated with hyperplane arrangements.
method Replaces entries of coboundary maps with q-analogues and analyzes the resulting structures. result The q-deformation can be a cochain complex under certain conditions and yields local system cohomology groups.