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.
Quotients of Gordian and H(2)-Gordian graphs are hyperbolic.
Study of Gordian graphs' behavior at infinity for various local moves.
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…
We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that overwhelmingly, the knot graphs are not Gromov hyperbolic, with the exception of a part…
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…
We show that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrarily high bridge distance.
We study the gordian graph of all knots in : 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.
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…
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…
Paper finds first examples of unlinked knots that can't be separated.
Lower bounds on Gordian distance using Blanchfield pairings.
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…
New bounds for knot distances using Khovanov homology.
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…
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 and unknotting number for any positive integer .
A knot is said to be Gordian adjacent to a knot if is an intermediate knot on an unknotting sequence of . We extend previous results on Gordian adjacency by showing sufficient conditions for Gordian adjacency between classes of positive braid knots through manipulations of braid words. In additio…
New examples of gordian unlinks show different rope geometries.
Study knots with genus one, finds Gordian distance and cosmetic crossing constraints.
Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in . Later Horiuchi and Ohyama defined Gordian complex of virtual knots using -move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing cha…
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…
New invariant measures how many twists are needed to unknot welded knots.
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…
Paper introduces a new invariant for planar knotoids.
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…
Given a link in 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…
We construct a pair of isotopic link configurations that are not thick isotopic while preserving total length.
The paper extends graph signatures to Klein graphs and foams, linking signatures to knot properties.
Unified proof of knot unknotting bounds using Ma-Qiu index.
Study on knot unknotting numbers and their behavior under connected sums.
New unknots with geometric constraints exist, proving a long-standing conjecture.
Algorithm simplifies Khovanov homology computations 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…
Defines a new Upsilon torsion function for knot Floer homology.
New methods for delta-moves on algebraically split links identified.
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.
Given a knot K in S^3, let u^-(K) (respectively, u^+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants l^-(K), l^+(K) and l(K), which give lower bounds on u^-(K), u^+(K) and the unknotting numb…
Geometric trick simplifies link homotopy and concordance.
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.
New examples show limits of physical link isotopies.
Proves links can be simplified to trivial form in few changes, limiting Milnor's invariants.
Line graph transformation aids graph isomorphism tests by excluding challenging graph properties.
Proposes MGMN for end-to-end graph similarity learning.
The paper explores graphons of line graphs from sparse finite graphs.
MxPool learns graph features from diverse graphs using a hierarchical structure.
Study the geometry of graph product extension graphs.
Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.